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THE

ROMANCE OF MATHEMATICS.

THE

Romance of Mathematics :

BEING

THE ORIGINAL RESEARCHES

OF

3^ Sabs ^xoftaaox 0f (iirtham doUege

IN

Polemical Science, with some Account of the Social

Properties of a Conic ; Equations to Brain

Waves; Social Forces ; and the Laws

of Political Motion,

BY

P. ^AMPSON, M.A.,

ORIEL COLLEGE, OXFORD.

LONDON :
ELLIOT STOCK, 62, PATERNOSTER ROW.

1886.

INTRODUCTION.

The lectures, essays, and other matter
contained in these pages have been dis-
covered recently in a well-worn desk which
was formerly the property of a Lady Pro-
fessor of Girtham College ; and as they
contain some original thoughts and in-
vestigations, they have been considered
worthy of publication.

How they came into the possession of
the present writer it is not his intention to
disclose ; but inasmuch as they seemed to
his unscientific mind to contain some im-
portant discoveries which might be useful to
the world, he determined to investigate
thoroughly the contents of the mysterious
desk, and make the public acquainted with
its profound treasures. He found some
documents which did not refer exactly to
the subject of * Polemical Mathematics;'
but knowing the truth of the Hindoo

vi Introdtiction,

proverb, *The words of the wise are
precious, and never to be disregarded,'
and feeling sure that this Lady Professor
of Girtham College was entitled to that
appellation, he ventured to include them
in this volume, and felt confident that in
so doing he would be carrying out the
intention of the Authoress, had she ex-
pressed any wishes on the subject. In
fact, as he valued the interests of the State
and his own peace of mind, he dared not
withhold any particle of that which he
conceived would confer a lasting benefit
on mankind.

Internal evidence seems to show that
the earlier portion of the MS. was written
during the period when the authoress was
still tn statu pupillari ; but her learning
was soon recognised by the Collegiate
Authorities,- and she was speedily elected
to a Professorship. Her lectures were
principally devoted to the abstruse subject
of Scientific Politics, and are worthy of the
attention of all those whose high duty it is
to regulate the affairs of the State.

The Editor has been able to gather from

Introduction. vii

the varied contents of the desk some
details of the Author's life, which increase
the interest which her words excite ; and
he ventures to hope that the public will
appreciate the wisdom which created such
a profound impression upon those whose
high privilege it was to hear the lectures
for the first time in the Hall of Girtham
College.

CONTENTS.

PAPER PAGE

I. Some Remarks on Female Edu-
cation :

Cambridge Man's Powers of Ap-
plication. — Torturing Ingenuity of
Examiners. — Slajring an Enemy. —

* Concentration.' — * Tangential Ac-
tion.* — 'Gravity' . . . . i

II. Lecture on the Theory of
Brain Waves and the Trans-
migration AND Potentiality
of Mental Forces ... 15

iiL The Social Properties of a Conic
Section, and the Theory of
Polemical Mathematics:
* Circle.'—* Parabola.'—' ElUpse.'

* Eccentricity of Curves ' . . 25

X Contents.

PAPER PAGE

IV. The Social Properties of a Conic
Section (continued) :

* Ellipse.' — Most favoured State. —
Alarming Result of Suppression of
House of Lords. — Analogies of
Nature. — Directrix. — Contact of
Curves and States. — * Hyperbola.* —
Problems. — Radical Axis and Pa-
triotism. — Extension of Franchise to
Women. — Correspondence . 39

V. Social Forces, with some Ac-
count OF Polemical Kinematics :
The Use of Imagination in Scien-
tific Discovery. — Kinetic and Poten-
tial Energy. — Social Statics and
D3mamics. — Attractive Forces. —
Cohesion. — Formation of States. —
Inertia. — Dr. Tyndall on Social
Forces ...... 71

VI. Social Forces {continued) : Pole-
mical Statics and Dynamics :

'Personal Equation.' — Public
Opinion, how calculated. — Impulsive
Forces. — Friction. — Progress . . 89

Contents. xi

PAPER PAGE

VII. Laws of Political Motion :

M. Auguste Comte on Political
Science. — First Law of Motion. —
The Biology of Politics. — Stages of
Growth and Decay of States. — Doc-
trine of Nationality. —Doctrine of
Independence. — Law of Morality. —
Ignorance of Electors and Selfishness
of Statesmen opposed to Action of
Law. — Final * Reign of Law ' . . loi
VIII. The Principle of Polemical Co-
hesion :

Centralization. — Co-operation of
States. — Marriage. — Trade Unions.
— International Law . . • "S
Extracts from the Diary of the

Lady Professor . . . .125
Conclusion '129

PAPER I.

SOME REMARKS OF A GIRTHAM GIRL ON
FEMALE EDUCATION.

[TTiis essay upon Female Education was evi-
dently written when the future Professor ofGirtham
College was still in the lowlier condition of student-
ships before she attained thai eminence for which
her talents so justly entitled her. Its unfinished
condition tends to show thai it was probably
evolved during moments of relaxation from severer
studies^ without any idea of subsequent publica-
tion.]

Oh, why should I be doomed to the
degradation of bearing such a foolish
appellation ! A Girtham Girl ! I suppose
we have to thank that fiend of invention
who is responsible for most of the titular
foibles and follies of mankind — artful Al-
literation. The two G\ people imagine,
run so well together ; and it is wonderful
that they do not append some other de-
lectable title, such as * The Gushing Girl of
Girtham,' or *The Glaring Girl of Glittering

2 The Romance of Mathematics.

Girtham.' O Alliteration ! Alliteration !
what crimes have been wrought in thy
name ! Little dost thou think of the mis-
chief thou hast done, flooding the world
with meaningless titles and absurd phrases.
How canst thou talk of * Lyrics of
Loneliness/ ' Soliloquies of Song,' * Pearls
of the Peerage'? Why dost thou stay
thine hand? We long for thee to en-
rich the world with * Dreams of a Dotard/
the * Dog Doctor's Daughters/ and other
kindred works. Exercise thine art on these
works of transcendent merit, but cease to
style thy humble, but rebellious, servant a
Girtham Girl !

But what's in a name ? Let the world's
tongue wag. I am a student, a hard-work-
ing, book-devouring, never-wearied student ,
who burns her midnight oil, and drinks the
strong bohea, to keep her awake during
the long hours of toil, like any Oxford or
Cambridge undergraduate. I often wonder
whether these mighty warriors in the lists
— the class lists, I mean — really work half
so hard as we poor unfortunate ' Girls of
Girtham.' Now that I am writing in strict

Remarks on Female Education. .3

confidence, so that not even the walls can
hear the scratchings of my pen, or under-
stand the meaning of all this scribbling, I
beg to state that I have my serious doubts
upon the subject; and when last I at-
tended a soiree of the Anthropological
Society, sounds issued forth from the
windows of the snug college rooms, which
could not be taken as evidences of pro-
found and undisturbed study.

Sometimes I glance at the examination
papers set for these hard-working students,
in order that they may attain the glorious
degree of B.A.', and astonish their sisters,
cousins, and aunts by the display of these
magic letters and all-resplendent hood.
And again I say in strict confidence that
if this same glorious hood does not adorn
the back of each individual son of Alma
Mater, he ought to be ashamed of him-
self, and not to fail to assume a certain
less dignified, but expressive, three-lettered
qualification. But before those Tripos
Papers I bow my head in humble adora-
tion. They sometimes take my breath
away even to read the terrible excruciating

4 The Romance of Mathematics.

things, which seem to turn one's brain
round and round, and contort the muscles
of one*s face, and stop the pulsation of
one's heart, when one tries to grasp the
horrid things.

Here is a fair example of the ingenuity
of the hard-hearted examiners, who re-
semble the inquisitors presiding over the
tortures of the rack, and giving the hate-
ful machine just one turn more by way of
bestowing a parting benediction on their
miserable victims :

* A uniform rod ' (it is a marvellous act
of mercy that the examiner invented it
uniform; it is strange that its thickness
did not vary in some complicated manner,
and become a veritable birch-rod !) * of
length 2Cy rests in stable equilibrium'
(stable ! another act of leniency !), * with
its lower end at the vertex of a cycloid
whose plane is vertical ' (why not incline
it at an angle of 30' ?) * and vertex down-
wards, and passes through a small, smooth,
fixed ring situated in the axis at a distance
b from the vertex. Show that if the equi-
librium be slightly disturbed, the rod will

Remarks on Female Edticatioit. 5

perform small oscillations with its lower
end on the arc of the cycloid in the time
4^y/^\ c-+3(6-c)' (, ^i^ere 2^ is the length

of the axis of the cycloid.*

A sweet pretty problem, truly! And
there are hundreds of the same kind —
birch-rods for every back! How the
examiner must have rejoiced when he
invented this diabolical rod, with its equi-
librium, its oscillations, its cycloid, and
other tormenting accessories. And yet, I
suppose, before my days of studentship
are over, I shall be called upon to attack
some such impregnable fortresses of mathe-
matics, when I hope to be declared equal
to some twentieth wrangler, if I escape
the misfortune of sharing a portion of the
* wooden spoon.'

Ah, you male sycophants ! You would
prevent us from competing with you ; you
would separate yourselves on your island
of knowledge, and sink the punt which
would bear us over to your privileged
shore. Of all the twaddle — forgive me,
male sycophants ! — that the world has ever

6 The Romance of Mathematics.

heard, I think the greatest is that which
you have talked about female education.
And the best of it is, you are so anxious
about our welfare ; you are so afraid that
we should injure our health by overmuch
mental exertion ; you profess to think that
our brains are not calculated to stand the
strain of continued mental exercise; you
think that competition is not good for th^
female mind ; that we are too competitive
by nature — too ambitious ! Yes, we are
so ambitious that we would enter the
lists with those who are asked in Public
Examinations to find the simple interest
on ;^i,ooo for 5 years at 6 J per cent. ;
so ambitious that we would compete with
those who are requested to disclose the
first aorist middle of ru^rw. Oh, think of
the mental strain involved in such ques-
tions ! How it must ruin your health to
find out how many times a wheel of radius
6 feet will turn round between York and
London, a distance of 200 miles ! It is
quite wonderful how your brains, my dear
male sycophants, can stand such fearful de-
mands upon your intelligence and industry!

Remarks oft Female Education. 7

But you are so kind to us, so afraid of
our health ! Really, we are much obliged
to you. If you married one of us, or
became our guardian, or left us a legacy,
we should then recognise your interest in
us, and be very grateful to you for your
good advice. But as matters standi we
are quite capable of taking care of our-
selves. We will promise not to work too
hard, if you will promise not to weary us
with your paternal jurisdiction.

But, male sycophants, I want a word
with you. Why do you object to our
taking degrees, or going in for examina*
tions in order to qualify ourselves for Our
duties in life ? You need not speak out
loud if you would rather not. Are you
not just a little afraid that we might eclipse
you? And it is not pleasant to be beaten
by a woman, is it ? And then you profess
to think that we ought to be all house^
wives and cooks, and knitters of stockings,
and sewers-OQ of our husbands' buttons ;
but what if we have no husbands, no
buttons to sew? And is it not a little
selfish, my dear male sycophant, to wish

8 The Romance of Mathematics.

to keep us all to yourself? to attend upon
the wants of the lords of creation, who
often distinguish themselves so much in
the domain of science?

Now, look me straight in the face (no
shirking, sir !). Is it not jealousy — ^green-
eyed, false-tongued jealousy — which saps
your generous instincts, and makes you
talk rubbish and nonsense about strains,
and brains, and ambition, and the like?
And if that is not hypocritical, I do not
know what is.

Well, good-day to you, male sycophant !
I really have not time to indulge myself in
scolding you any more. You are a good
creature, no doubt; and when you have
shown us what you can do, and can esti-
mate the capacity of the female brain, and
take a common-sense view of things, we
will recognise your privilege to speak;
and when I am the presiding genius of
Girtham College, I will grant you the use
of our hall for the purpose of lecturing to us
on * Women's Rights,' or, as you may prefer
to entitle your discourse, * Men's Wrongs.*

Remarks on Female Education, g

Oh, this is shameful ! I really am very
sorry. Here have I been wasting a good
half-hour in dreaming, and slaying an
imaginary enemy with envenomed words
and frequent dabs of ink. If I cannot
concentrate my mind more on these
mathematical researches, I fear a dreadful
* plough' will harrow my feelings at the end
of my sojourn in these halls of learning.

Concentration! How many of our
words and ideas and thoughts are derived
from that primal fount of all arts and
sciences — mathematics ! Here is one
which owes its origin to the mathema-
tically trained mind of some early philo-
logical professor, who had learnt to apply
his scientific knowledge to the enrichment
of his native tongue. He quoted to
himself the words of the Roman poet :

' Ego cur, acquirere pauca
Si possum, invideor, cum lingua Catonis et Ennt
Sermonem.patrium ditaverit, et nova rerum
Nomina protulerit ? Licuit, semperque licebit.'

His mind conceived endless figures of
circles and ellipses scattered promiscuously
over the page, defying the attempts of the

10 The Romance of Mathematics.

student to reduce them to order. What
must he do before he can apply his
formulae and equations, determine their
areas, or describe their eccentric motion ?
He must reduce them to a common
centre, and then he can proceed to
calculate the abstruse problems in con-
nection with the figures described. They
may be the complex motions of double-
star orbits, or the results of the impact of
various projectiles on the tranquil surface
of a pool. It matters not — the principle
is the same; he must concentrate, and
reduce to a common centre.

This is the great defect of those who
have no accurate mathematical knowledge;
they cannot concentrate their minds with
the same degree of intensity upon the
work which lies before them* Their
thoughts fly off at a tangent, as mine do
very often; but then I have not been
classed yet in the Tripos; and, O male
poetical sycophant, you niay be right after
all when you say:

*" O woman ! in our hours of ease . ,

Uncertain, coy and hard to please,
As variable as the noon-day shade.'

Remarks on Female Education, ii

Yes/ as variable as the most variable
quantities x,y^ z. I, a student of Girtham
College, blush to own that my thoughts
very often fly off* at a tangent.

* Fly off* at a tangent T All hail to
thee, most noble mathematical phrase I
Here is another fine mathematical ex-
pression, plainly exemplifying the action
of centrifugal force. The faster the
wheel turns, the greater is the velocity of
the discarded particles which fly off* along
the line, perpendicular to the radius of
the circle. The world travels very fast
now ; the increased velocity of the transit
of earthly bodies, the rate at which they
live, the multiplicity of engagements,
etc, have made the social world revolve
so fast that the speed would have startled
the torpid life of the last century. And
what is the result ? Men's thoughts fly off
at a tangent ; they are unable to concen-
trate their minds on any given subject;
they are content with hasty generalisms,
with short magazine articles on important
subjects, which really require large volumes
and patient ^tudy to elucidate them fully. .

1 2 The Romaftce of Mathematics.

What we want to do is to increase the
attractive force, in order to prevent this
tangential motion — to increase the force of
gravity.

* Well/ says the young lady who loves
to revel in the 'Ghastly Secret of the
Moated Dungeon,' or the * Mysteries of
Footlight Fancy,' *you are grave enough.
Pray don't increase your gravity !'

Thank you, gentle critic. I will, in
turn, ask you one favour. Leave for once
the 'Mysteries of Footlight Fancy;' seek
to know no more 'ghastly secrets,' and
increase your gravity — your mental
weight ; and hence your attraction in the
eyes of all who are worth attracting will be
marvellously increased, by understanding
a little about Newton's law of universa
gravitation, and don't fly off at a tangent

At the end of this portion of the MS.
the editor of these papers discovered a
photograph which, from subsequent in-
quiry, proved to be that of the accom-
plished authoress of the above reflections

Remarks on Female Education. 13

The face is one of considerable beauty,
with eyes as clear, steadfast, and open as
the day. There is a degree of firmness
about the mouth, but it is a sweet and
pretty one notwithstanding ; and a smile,
half scornful, half playful, can be detected
lurking about the corners of the lips,
which do not seem altogether fitted for
pronouncing hard mathematical terms and
abstruse scientific problems. This photo-
graph might have been the identical one
which nearly brought an enamoured youth
into grave difficulties by its secretion in
the folds of his blotting-paper during
examination. The said enamoured youth
bad evidently placed it there for the sake
of its inspiring qualities ; and it was said
that all his hopes of gaining the hand of
the fair original depended upon his pass-
ing that same examination. But the
wakeful eye of a stern examiner had
watched him as he turned again and
again to consult the sweet face which
beamed from beneath his blotting-paper;
and he narrowly escaped expulsion from
the Senate-house on the charge of * crib-

14 The Romance of Mathematics.

bing.' Certainly he took a mean advan-
tage of his fellow-sufferers, if this were the
identical photograph, for it portrays a
most inspiring face. Forgive us, lenient
reader ; one moment ! There — thank you
— we have done. And now we will proceed
to disclose the researches and original
problems which the MS. contains.

Evidently the collegiate authorities were
not slow in recognising the talents of the
assiduous student, and elected her with-
out much delay to a Professorship of
Girtham. In this capacity the learned
lady delivered several lectures, of which
the second MS. contains the first of the
eries.

PAPER II,

lecture on the theory of brain waves
and the transmigration and poten-
tiality of mental forces.

Professors and Students of the
University of Girtham, my Lords,
Ladies, and Gentlemen, — I have the
honour to bring before you this evening
some original conceptions and discoveries
which have been formulated by me during
my researches in the boundless field of
mathematical knowledge; and though you
may be inclined at first to pronounce
them as somewhat hastily conceived hypo-
theses, I hope to be able to demonstrate
the actual truth of the propositions which
I shall now endeavour to enunciate. It
is with some feelings of diffidence that I
stand before so august an assembly as the
present ; and if I were not actually con:

i6 The Romance of Mathematics,

vinced of the accuracy of my calculations,
I should never have presumed to appear
before you in the character of a lec-
turer. But ^Mag7ia est Veritas^ et ^rceba-
lebit* I cast aside my maiden timidity; I
clothe myself in the professorial robe which
you have bestowed upon me, and sacrifice
my own feelings on the altar of Truth.

I have been engaged, as you are doubt-
less aware, for some years in the pursuit
of mathematical research, exploring the
mines of science, which have of late been
worked very persistently, but often, like
the black diamond mines, at a loss. Con-
currently with these researches, I have
speculated on the great social problems
which perplex the minds of men, both
individually and collectively. And I have
come to the conclusion that the same
laws hold good in both spheres of work ;
that methods of mathematical procedure
are applicable to the grand social prob-
lems of the day and to the regulation
of the mutual relations which exist
between man and man. Take, for
example, the Force of public opinion. Of

The Theory of Brain Waves. 17

what is it composed ? It is the Resultant
of all the forces which act upon that
which is generally designated the ' Social
System.' Public opinion is a compromise
between the many elements which make
up human society; and compromise is a
purely mechanical affair, based on the
principle of the Parallelogram of Forces.
Sometimes disturbing forces exert their
influence upon the action of Public
Opinion, causing the system to swerve
from its original course, and precipitating
society into a course of conduct incon-
sistent with its former behaviour ; and it
is the duty of the Governing Body to
eliminate as far as possible such disturb-
ing forces, in order that society may
pursue the even tenor of its way.

Professors, we have one great problem
to solve; and all questions social, poli-
tical, scientific, or otherwise, are only
fragments of that great problem. All
truths are but different aspects of different
applications of one and the same truth ;
and although they may appear opposed,
they are not really so ; and resemble

i8 The Romance of Mathematics.

lines which run in various directions, but
lovingly meet in one centre.

Now, let us take for our consideration
the secret influence which men exert upon
each other, apart from that produced by
the power of speech (although that would
come under the same general law). As
mathematicians, you are aware that the
undulatory theory of light and heat and
sound are now accepted by scientific men
as the only sure basis of accurate calcula-
tion. We know that the rays of light
travel in waves, and the equation repre-
senting the waves is

a , 2v
>'= -sm - {vt-r\

where y is the disturbance of the ether,
a the initial amplitude, r the distance
from the starting-point, X the wave-length,
and V the velocity of light. Sound and
heat likewise have much the same form
of equation. Now, I maintain that the
waves of thought are governed by the
same laws, and can be determined by
an equation of the same form. You are

The Theory of Brain Waves. 19

aware that in all these equations a certain
quantity denoted by X appears, and varies
for the different media through which the
sound, or light, or heat passes, and which
must be determined by experiment Now,
in my equation for brain waves, the same
quantity X appears, which must be deter-
mined by the same method — by expert-
ment But how is this to be done ? After
mature deliberation and much careful
thought, I have discovered the method for
finding X. This method is mesmerism.
We find the ratio of brain to brain — the
relative strength which one bears to
another; and then by an application of
our formula we can actually determine the
wave of thought, and read the minds of
our fellow-creatures. An unbounded field
for reflection and speculation is here sug-
gested. Like all great discoveries, the
elements of the problem have uncon-
sciously been utilized by many who are
unable to account for their method of
procedure. For example, thought-readers,
mesmerists, and the like, have uncon-
sciously been working on this principle^

2 — 2

20 The Romance of Mathematics,

although lack of mathematical training
has prevented them from fully mastering
the details of the problem. Hence in
popular minds a kind of mystery has
hung about the actions of such people,
and excited the curiosity of mankind.

The development of this theory of brain
waves may be of great practical utility to
the world, f It shows that great care ought
to be exercised in the domain of thought,
as well as that of speech. For example :
A man has made a startling discovery,
from which he expects to receive consider-
able worldly advantage. He would be
careful not to disclose his discovery in
speech to his acquaintances until his plans
are sufficiently matured, lest they should
impart it to the world, patent his device,
and reap the reward. But while he is
endeavouring to talk carelessly about it,
the wave of thought may be travelling
from brain to brain, suggesting the exist-
ence of the discovery; and if the con-
ditions are favourable, and \ sufficiently
small, it is possible that the idea itself
may be conveyed. Of course the more

The Theory of Brain Waves. 21

complicated the discovery, the less likely
would the wave convey the conceptioiL
Or suppose that one of the learned pro-
fessorial body of our sister university
should conceive an attachment for a lady-
student of Girtham College (of course a
very improbable supposition!), and the
infatuated savant became somewhat jealous
of another learned lecturer of the same
college (another improbability !), the fact
of his jealousy would be imparted to the
latter by a wave of thought, and might
cause considerable confusion in the serene
course of love or science. The fact of
the existence of the wave is indisputable.
What do all the stories of impressions and
double-sight teach us? How could the
intelligence of the death of Professor
Steele have been conveyed to his friend
and fellow-student, Professor Tait — the
one at Cambridge, the other at Edinburgh
— ^were it not for the existence of some
wave, which, like that of electlicity, wings
its rapid flight unobserved by human eyes?
Are all the records of the Psychical Society
only myths and legends bred of super-

22 The Romance of Mathematics.

stitious fancy? It were hard to sup-
pose so.

But if, gentlemen, and ladies especially,
you wish to keep your secret discoveries
to yourselves, watch over your thoughts as
well as your words; for my researches
prove, and the universal experience of
mankind corroborates the fact, that some
portion of your inmost thoughts and secret
desires are understood by your neighbours
(especially when X is small!); that they
travel along the waves which I have at-
tempted to indicate; and if you would
desire to extend your influence in the
world, probe the secret instincts of man-
kind, and prevent yourself from being
deceived and wronged — study the art and
science of Brain Waves.

The following verses of rather doubtful
merit were found in connection with the
previous MS. They were evidently written
by a different hand ; but inasmuch as they
were deemed worthy of preservation by

The Theory of Brain Wave$. 23

the learned owner of the sealed desk, we
venture to publish them. They are closely
connected with the previous lecture, and
were evidently composed by an admirer
of the fair lecturer who did not share her
love for scientific research.

Wavelet,* wing thy airy flight ;

Let thine amplitude be great ;
Tell her all my thoughts to-night|

How I long to know my fate.

All the fields of Mathematics
I have roamed at her decree ;

From Binomial and Quadratics,
To the strange hyperbole, f

I have soared through Differential,
Deeply drunk of Finite Boole ;t

Though its breath is pestilential.
Reeking of the hateful School.

I have tried to shape a Conic,

Vainly read the Calculus ;
But my feebleness is chronic.

Morbus Matheniaticus.

* We presume this is addressed to an imaginary
brain wave.

t We observe here the dash of an indignant pen,
and a substituted for e. But now the rhyme is
spoiled. Gentle Muse, thou art sacrificed by the
stem hand of Mathematical Truth !

X Query : Does the writer refer to the learned
treatise on Finite Differences by Professor Boole ?

24 The Romance of Mathematics.

All my curves are cardioidal ;

I confuse my x and y%
Which they say is suicidal ;

And my tutor vainly sighs.

Wivelet, tell her how I love her,
As she mounts her learned throne ;

And that love I hope insnr cover
AH the failings which I own.

Wavelet, cry to her for pity ;

Bid her end this bitter woe ;
I might do something 'in the city/

But never pass my Little-go.

PAPER III.

lecture on the social properties of
a conic section, and the theory of
polemical mathematics.

Most Learned Professors and Stu-
dents OF THIS University, — From the
interest manifested in my first lecture, I
conclude that my method of investigation
has not proved altogether unsatisfactory
to you, and I hope ere long to produce
certain investigations which will probably
startle you, and revolutionize the current
thought of the age. The application of
mathematics to the study of Social
Science and Political Government has
curiously enough escaped the attention of
those who ought to be most conversant
with these matters. I shall endeavour to
prove in the present lecture that the
relations between individuals and the
Government are similar to those which

26 The Romance of Mathematics.

mathematical knowledge would lead us
to postulate, and to explain on scientific
principles the various convulsions which
sometimes agitate the social and political
world.

Indeed, by this method we shall be able
to prophesy the future of states and
nations, having given certain functions and
peculiarities appertaining to them, just as
easily as we can foretell the exact day and
hour of an eclipse of the moon or sun. In
order to do this, we must first determine
the social properties of a conic section.

For the benefit of the unlearned and
ignorant, I will first state that a cone is
a solid figure described by the revolution
of a right-angled triangle about one of the
sides containing the right angle, which
remains fixed. The fixed side is called
the axis of the cone. Conic sections are
obtained by cutting the cone by planes.
It may easily be proved that if the angle
between the cutting plane and the axis be
equal ^o the angle between the axis and
the revolving side of the triangle which
generates the cone, the section described

Social Properties of a Conic Section. 27

on the surface of the cone is a parabola;
if the former angle be greater than the
latter, the curve will be an ellipse ; and if
less, the section will be a hyperbola.

But the simplest conic section is, of
course, a circle, which is formed by a plane
at right angles to the axis of the cone;
and the simplest circle is that formed
by a plane passing through the apex of the
cone. All this is simple mathematics;
and let beginners consult more elementary
treatises than this one to satisfy them-
selves on these points. But if they will
assume these things to be true, they will
know quite enough for our present
purpose. The simplest conic section
of all has been proved to be a ^in/.
Now, this represents the simplest and
original form of society, a single family.
'It is not good for man to be alone'
was the first observation made by the
wise Creator upon the rational crea-
ture whom He had introduced into
Paradise as its lord. Marriage is the
rudiment of all social life, from which
all others spring, out of which all others

28 The Romance of Mathematics.

are developed. Around the parents' knees
soon cluster a group of children, and in
their relation to each other we discern the
earliest forms of law and discipline — the
bonds by which society is held together.
When the children grow up, separate
households are formed; and then the
multiplication of families, the congregating
of men together for purposes of security
and mutual advantages in division of
labour; and thus is gradually formed a
state, which is only the development of
the family — the king representing the
parent, and ruling on the same principle.
Mathematically speaking, our plane no
longer passes through the apex. The
point represented the single family;
but keeping the plane horizontal, we
move it along the axis, the sections
will become circles^ which represent
mathematically the next simplest form of
society, where the centre is the seat of
government, which is connected with each
individual member of the social circle by
equal radii. The social property of a
circle is that of a monarchical government

Social Properties of a Conic Section. 29

in its purest and simplest form. The
larger the circle becomes {ie.^ the further
you move the plane from the apex), the
greater the distance between the individual
and the monarch. Therefore, the more
independent the monarchy becomes, and
the less influence do individuals possess
over the ruling power. Hence, we may
infer that as years roll on, the government
will become more despotic; but the
stability of the country diminished, and
probably some individual particle, when
sufficiently withdrawn from the attraction
of the central head, will begin to revolve
on its own account, and spontaneously
generate a government of its own. We
may, therefore, conclude from mathema-
tical reasoning that an unlimited monarchy,
though advantageous for small states, is
not a safe form of government for a large
or populous country, inasmuch as the
people do not derive much benefit from
the sovereign; the mutual attiaclion,
which ought to exist in a flourishing state
between the ruler and the ruled, is
weakened; and the isolation of the

30 The Romance of Mathematics.

monarch tends to make him still more
despotic. As a practical example of the
truth of the foregoing statement, I may
mention the present condition of Russia,
which shows that the result of an un-
limited monarchy, in a large and unwieldy
social circle, is such as we should have
reasonably expected from mathematical
investigations.

Invariably, under the circumstances
which I have described, the country will
become disorganized; the sovereign will
cease to have any power over the people,
and the country will become a chaos,
without order, influence, or power.

When the centre of a conic section
moves along the axis of the curve to
infinity, banished by the mutual consent
of the individual particles which compose
the curve, or the nation, a figure is formed,
called a parabola. This is the curve
which the most erratic bodies in the
universe describe in space, as they rush
along at a speed inconceivable to human
minds, and are supposed to produce all
kinds of mischief and injury to the

Social Properties of a Conic Section, 31

worlds whose courses they wend their way
among.

This curve, then, represents the position
which the nation assumes when the con-
stituted monarchy, the centre of the system,
has been banished to infinity, A revolution
has occurred; the monarch has been
dethroned ; and it is not hard to see that
the same erratic course which the comet
pursues in its flight, is observable with
respect to the social system which is
represented by a parabola. We observe
with eager scrutiny the wanderings of
these erratic comets. They appear sud-
denly with their vapoury tails ; sometimes
they shine upon us with their soft, silvery
light, brilliant as another moon; some-
times they stand afar off in the distant
skies, and deign not to approach our
steady-going earth, which pursues its
regular course day by day, and year by
year. Then, after a few days' coy in-
spection of our planet from different
points of view, they fly to other remote
parts of the universe, and do not con-
descend to show themselves again for a

32 The Romance of Mathematics.

hundred years or so. Such is the erratic
conduct of a heavenly body whose course
is regulated by a parabolic curve.

We may look for similar eccentric be-
haviour on the part of a community,
nation, or state, whose centre is at in
finity, whose constitution has been vio-
lently disturbed, and whose monarchy is
situated in the far-ofif regions of unlimited
space. The erratic course of Republican
rule is proverbial There is no stability,
no regularity. To-day we may observe its
brilliancy, which seems to laugh at and
eclipse the sombre shining of more
steady and enduring worlds; but ere to-
morrow's moon has risen, it may have
vanished into the regions of eternal night,
and we look for its bright shining light in
the councils of the nations, but it has
ceased to shed its rays, and we are dis-
appointed. Sometimes it is asked, with
fear and trembling : * What would be the
effect if our earth were to come in contact
with the tail of a comet ? Should we be
destroyed by the collision, and our pon-
derous world cease to be?' But we are

Social Properties of a Conic Section. 33

assured that no such disastrous results
would follow. We have already passed
through the tails of many comets, but we
have not discovered any inconvenient
change in our ordinary mode of pro-
cedure. It is probable that the comet's
tail is composed of no solid substance.

We may therefore infer by analogy that
a Republican State would not offer any
powerful resistance if it were to come into
collision with a nation possessing a more
settled form of government A shower of
meteoric stones, like passing fireworks,
might take place; but beyond that
nothing would occur to excite the fear,
or arouse the energies of the more
favoured nation. As an example of the
weakness of a Republican State I may
mention France. There we see an in-
dustrious race of people, endowed with
many natural gifts and graces, a country
rich and productive; and yet, owing to
the unsettled nature of its government, all
these natural advantages are neutralized ;
its course amongst the nations is erratic
in the extreme, a spectacle of feeble ad^

3

34 The Romance of Mathematics.

ministration ; and it would offer no more
resistance to a colliding Power than the
empty vacuum of a comet's tail. This
example will demonstrate to you the truth
of our theory with regard to the instability
of a social system which is geometrically
represented by a parabolic curve.

We will now turn from this picture of
insecurity and unrest to another figure
which possesses most advantageous social
properties. I refer to the ellipse. An
ellipse is a curve formed by the section
of a cone by a plane surface inclined at
an angle to the vertical axis of the cone,
greater than the angle between the axis
and the generating line.

Now, this is a curve which possesses
most attractive properties. It is the curve
which the earth and other planetary orbs
describe around the centre of the solar
S3rstem, as if nature intended that we
should take this figure as a guide in
choosing the most advantageous social
system. It possesses a centre, C, in view
of all the particles which compose the
curve, and connected with them by close

Social Properties of a Conic Section, 35

ties. It has two foci, S and S', fixed
points, by the aid of which we may trace
the curve.

In the interpretation of this figure, the
centre of the curve represents the throne
of monarchy. There is no tendency here
to revolutionize the State, to banish the
ruling power, and institute a Republican
form of government ; but inasmuch as we
saw the weakness of an absolute monarchy
in large and populous States, as repre-
sented by the circle, the wisdom of an
elliptical social system has ordained that
there shall be two foci, or houses of
representatives of the people, who shall
assist in regulating the progress of the
nation. Here we have a limited monarchy ;
the throne is supported by the representa-
tives of the people ; and the nearer these
foci of the nation are to the centre (/^.,
in mathematical language, the less the
eccentricity of the curve), the more per-
fect the system becomes — the greater the
happiness of the community.

In cases where the eccentricity becomes
yery great, the beauty of the curve is

3—2

36 The Romance of Mathematics.

destroyed, and ultimately the ellipse is
merged into one straight line. Most
learned Professors, here we have a terrible
warning of the awful result of too much
eccentricity. Whether we regard the life
of the nation or of the individual, let all
bear in mind this alarming fact, that
eccentricity of thought, habit, or behaviour
may result, as in the case of this unfor-
tunate ellipse, which once presented such
fair and promising proportions to the
student's admiring gaze, in the 'sinister
effacement of a man,' or the gradual ab-
sorption of a State into an uninteresting
thing 'which lies evenly between its ex-
treme points.'

The great examples of Bacon, of Mil-
ton, of Newton, of Locke, and of others,
happen to be directly opposed to the
popular inference that eccentricity and
thoughtlessness of conduct are the neces-
sary accompaniments of talent, and the
sure indications of genius. I am indebted
to Lacon for that reflection. You may
point to Byron, or Savage, or Rousseau,
and say, * Were not these eccentric people

Social Properties of a Conic Section, 37

talented?' 'Certainly,' I answer; *but
would they not have been better and
greater men if they had been less eccen-
tric — if they had restrained their caprice,
and controlled their passions? Do not
imagine, my young students of this uni-
versity, that by being eccentric you will
therefore become great men and women
of genius. The world will not give you
credit for being brilliant because you affect
the extravagances which sometimes ac-
company genius. Some of you ladies, I
perceive, have adopted a peculiar form of
dress, half male, half female ; or, to be
more correct, three-fourths male, and one-
fourth female. Do not imagine that you
will thus attain to the highest honours in
this university by your eccentricity, unless
your talents are hid beneath your short-cut
hair, and brains are working hard under
your college head-gear. As well might we
expect to find that all females who wear
sage-green and extravagant aesthetic cos-
tumes are really born artists and future
Royal Academicians. It is apparent that
many aspirers to fame and talent are eager

38 The Romance of Mathematics.

to exhibit their eccentricities to the gaze
of the world, in order that they may per-
suade the multitude that they possess the
genius of which eccentricity is falsely sup-
posed to be the outward sign.

I may remark in passing that the eccen-
tricity of a parabolic curve is always unity.
What does this prove ? You will remember
that a Republican State is represented by
a parabola. Therefore, however such a
nation may strive to alter its condition,
and secure a settled form of government,
its eccentricity will always remain the
same. It will always be erratic, peculiar,
unsettled; and this conclusion substan-
tiates our previous proposition with regard
to the condition of a social system repre-
sented by a parabola.

With regard to other advantages afforded
by an elliptical social system, we will de-
fer the consideration of this important
subject until my next lecture.

PAPER IV.

THE SOCIAL PROPERTIES OF A CONIC
SECTION, AND THE THEORY OF POLE-
MICAL MATHEMATICS — (continued).

Most learned Professors and Students
OF this University, — ^You have already
gathered from my preceding lecture my
method of procedure in the investigation
of the corresponding properties of curves
and States. You have perceived that we
have here the elements of a new science,
which may be extended indefinitely, and
applied to the various departments of self-
government and State control. This new
science of polemical mathematics is in
itself an extension of the principle of con-
tinuity^ for the discovery of which Poncelet
is so justly renowned. We can prove by
geometry that the properties of one figure
may be derived from those of another

40 The Romance of Mathematics.

which corresponds to it; and the new
science teaches us that if we can represent,
by projection or otherwise, a society of
particles or individuals on a plane surface,
the properties of the State so represented
are analogous to the properties of the
curve with which it corresponds. It is
only possible for me to touch upon the
elements of the science in these lectures,
but I hope to arouse an interest in these
somewhat unusual complications and
curious problems, that you may hereafter
make further discoveries in this unexplored
region of knowledge, and that the world
may reap the benefit of your labours and
abstruse studies. I have already, in my
previous lecture, touched upon the social
properties of the parabola, and examined
the constitution of erratic curves and
eccentric nations. It is my intention
to-day to speak of similar problems
which arise with reference to elliptical
States.

But, first, let me answer an objection
which may have occurred to your minds.

Social Properties of a Conic Section, 41

Am I wrong in my calculations in attri-
buting too much to the power and useful-
ness of forms of government ? Does the
well-being and happiness of a nation de-
pend on the government, or upon the
individuals who compose the nation?
Most assuredly, I assert, they rest upon
the former. Men love their country when
the good of every particular man is com-
prehended in the public prosperity ; they
undertake hazard and labour for the
government when it is justly administered.
When the welfare of every citizen is the
care of the ruling power, men do not
spare their persons or their purses for the
sake of their country and the support of
their sovereign. But where selfish aims
are manifest in Court or Parliament, the
people care not for State officials who are
indifferent to their country's weal; they
become selfish too ; Liberty hides her head,
and shakes off the dust of her feet ere she
leaves that doomed land, and the stability,
welfare, and prosperity of that country
cease.

42 The Romance of Mathematics,

I might refer you to many a stained
page of national history in order to prove
this. Compare the closing chapters of the
life of the Roman empire with the record
of the brave deeds of its ancient warriors
and valorous statesmen. Grecian pre-
eminence and virtue died when liberty
expired. I agree with Sidney when he
writes that it is absurd to impute this to
the change of times; for time changes
nothing, and nothing was changed in
those times but the government, and that
changed all things. These are his words :
' As a man begets a man, and a beast a
beast, that society of men which consti-
tutes a government upon the foundation
of justice, virtue, and the common good,
will always have men to promote those
ends; and that which intends the ad-
vancement of one man's desires and
vanity will abound in those that will
foment them.' I may not, therefore, be
altogether wrong in attributing the pros-
perity and well-being of a nation to the
form of government which it possesses.

Social Properties of a Conic Section. 43

We will now proceed to the considera-
tion of the social advantages which an
elliptical State affords. This is the form
of government and social position which
we, as a nation, at present enjoy; and
from mathematical considerations I am
of opinion that it is the best, and hope
that no change will ever be made in our
constitution. You may remember that I
have previously stated that an ellipse has
a centre and two foci, in view of all the
particles which compose the curve, and
connected with them by close ties. The
centre, in the projected figure, represents
the monarchy, which is limited ; and the
government is carried on by the aid of
the two houses of representatives of the
people, depicted in the projection by the
two foci.

Now the social advantages of the ellipse
are given by the fact that the sum of the
distances of any point from the foci is
always constant No particle is left out
in the cold ; no one does not possess
the advantages of a social government

44 ^^^^ Romance of Mathematics.

Though his distance may be far from the
Upper House, he has the advantage of
nearness to the Lower, and vice versa. The
sum of the distances is constant. The
extinction of one focus, the House of
Lords, for example, would create a com-
plete disorganization of the whole system :
the other focus would set up a powerful
magnetic attraction, and a curious bulb-
shaped curve would be evolved, very
different from the beautiful symmetrical
form which the original figure presented
to the eye. The centre of the system
would be disturbed ; and it is probable
that ere long it would disappear along the
axis and be vanished to infinity. Thus
the curve would become a parabola. This
is the alarming result of the extinction of
one focus. Abolish the House of Lords,
and you will soon find that the Throne
will be disturbed ; the State will become
disorganized ; the nation will become
confused by the magnetic force of the
Lower House, uncounteracted by any other
attraction ; and very soon a complete

Social Properties of a Conic Section. 45

revolution of the whole system will set
in : the monarch will be dethroned, and
a Republican form of government, with all
the eccentricities of a parabolic course,
will take the place of a more orderly and
settled constitution. This is a plain
deduction from our mathematical investi-
gations ; and it behoves all our statesmen,
our philosophers and great men, our
fellow-citizens and the humblest artisans
in our manufacturing towns, to weigh well
this alarming result of the abolition of
that House which has been threatened
with destruction ; and to ascertain for
themselves the truths upon which my pro-
positipn and reasoning rest.

I have already observed that the fact
that the earth's orbit and that of other
planets are in the form of ellipses ; that
the curvature of the earth is nearly the
same, ought to guide us in choosing this
particular curve as a model of the projec-
tion of a complete and most advantageous
social system.

The circle described on the major axis of

44 The Romance of Mathematics.

Though his distance may be far from the
Upper House, he has the advantage of
nearness to the Lower, and vice versa. The
sum of the distances is constant The
extinction of one focus, the House of
Lords, for example, would create a com-
plete disorganization of the whole system :
the other focus would set up a powerful
magnetic attraction, and a curious bulh-
shaped curve would be evolved, very
different from the beautiful symmetrical
form which the original figure presented
to the eye. The centre of the system
would be disturbed ; and it is probable
that ere long it would disappear along the
axis and be vanished to infinity. Thus
the curve would become a parabola. This
is the alarming result of the extinction of
one focus. Abolish the House of Lords,
and you will soon find that the Throne
will be disturbed ; the State will become
disorganized ; the nation will become
confused by the magnetic force of the
Lower House, uncounteracted by any other
attraction ; and very soon a complete

Social Properties of a Conic Section, 45

revolution of the whole system will set
in : the monarch will be dethroned, and
a Republican form of government, with all
the eccentricities of a parabolic course,
will take the place of a more orderly and
settled constitution. This is a plain
deduction from our mathematical investi-
gations ; and it behoves all our statesmen,
our philosophers and great men, our
fellow-citizens and the humblest artisans
in our manufacturing towns, to weigh well
this alarming result of the abolition of
that House which has been threatened
with destruction; and to ascertain for
themselves the truths upon which my pro-
positipn and reasoning rest.

I have already observed that the fact
that the earth's orbit and that of other
planets are in the form of ellipses ; that
the curvature of the earth is nearly the
same, ought to guide us in choosing this
particular curve as a model of the projec-
tion of a complete and most advantageous
social system.

The circle described on the major axis of

46 The Roinance of Mathematics.

an ellipse, is called the auxiliary circle^ and
affords much assistance in the investiga-
tion of the properties of an ellipse. As
we have already shown, the circle repre-
sents the simplest form of monarchical
government. Hence, if we compare the
form of government represented by an
ellipse (/.<?., such as we now enjoy) with
that of a system where the king is the only
governing power, we may obtain great
assistance in solving complicated political
problems.

In all conies there is a straight line
called the ' directrix,* which represents in
social or polemical science the laws of the
nation, and plays a prominent part in the
mutual relations of the individual particles.
For instance, in the case of the parabola,
the distance of any particle from the
directrix is equal to its distance from the
focus.

From this we may conclude that if an
individual deviates at all from the path
which the laws (or, directrix) indicate, if
he does not show true respect to the

Social Properties of a Conic Section. 47

decrees of the focal government, and
preserve the true position between them,
directly he is found deviating from his
course, he is quickly banished to a less
enlightened sphere. In an ellipse there
is less likelihood of his straying away from
the course which the directrix points out,
on account of the two-fold guidance which
he receives from the two focL

The following curious problem may be
noticed. If a parabola roll on another
parabola, their virtices coinciding, the
focus of the first traces out the directrix
of the second.

Here we come to the consideration of
the international relationship of States.
Two nations have the same form of
government (in this example this form is
Republican) ; their policies coincide : we
may conclude from this proposition that
the course which the government of one
nation will pursue, will be that which is
prescribed by the laws of the other.

The subject of the contact of curves
presents many interesting problems with

48 The Rofftance of Mathematics,

reference lo Polemical Science, and may
be extended indefinitely. It is well known
that there are diflferent orders of contact,
which are designated as the Jirsf, second,
or Mrd order. This last order may be
termed the * marriage of curves,' cemented
by the osculating circle, or 'wedding-
ring ;' and when two nations have contact
of the third order, they have formed a
very close alliance, and by calculation we
can obtain the radius of curvature, or size
of the wedding-ring, by means of which
they may be united.

The theory and nature of contact con-
stitute a branch of our newly discovered
science which we commend to the careful
consideration of those who have under-
taken the difficult and perplexing study of
international law. Alas ! too many States
refuse this friendly contact, and, con-
sequently, cut each other, instead of
blending in sweet accord. Their peace is
at best an armed neutrality; and if they
have contact of only the first or second
order, we can prove mathematically that

Social Properties of a Conic Section. 49

they are sure to intersect in some other
point or points ; and divergence of policy
and disturbed relations are the results.
Contact of the fAird, or highest^ order is
the only safe position for two allied, or
contiguous, States.

With your permission I will add a few
words to those I have already uttered
with regard to the directrix. As necessary
as the directrix is to the curve, so are
the corresponding laws to the State. I
will prove this fact by a few examples.
English people have laws, and know how
to obey them; therefore their numbers
increase ; they thrive and are prosperous.
A friendly critic of another nation has
said that the reason why Englishmen
rule the. world, is because they know how
to obey. On the other hand, the gipsies
have no laws; hence they become fewer
and less powerful. What is the condition
of all tribes and nations which are not
governed by laws? They invariably
remain poor and miserable. They are in
want of a directrix ; and if we could sup-

4

50 The Romance of Mathematics.

plement the gift with foci and centre, they
would soon emerge from their savage con-
dition, and become more civilized.

I have omitted to mention the hj^er-
bolic form of government The curve
formed by the intersection of the surface
of a cone with a plane will be a hyper-
bola, when the inclination of the cutting
plane to the axis of the cone is less than
the constant angle which the generating
line forms with the axis. It is manifest
that the plane will thus intersect the
higher cone, and produce the figure
which is known to mathematicians as
the hyperbola.

We may hence deduce the following
property of the corresponding hyperbolic
State. We take cognizance of that higher
cone with which the mundane affairs of
the lower cone are closely connected. As
an example of this system we may mention
the vast temporal rule and power of the
Papal Throne, which formerly exercised
such marvellous sway over the nations of
Europe. By an appeal to a Higher

Social Properties of a Conic Section. 51

Authority than that of earthly kings and
potentates was this rule exercised ; but its
hyperbolic form is fast passing away, and
degenerating into that of a circle with
indefinitely small radius. We shall not,
therefore, discuss the complex polemi-
cal problems which a hyperbolic State
suggests.

I will now mention a few problems
which are easily capable of proof, and
deduce from them the necessary conclu-
sions which must follow when we apply
our newly discovered principles of polemi-
cal science.

I. ' If from any point in a straight line
a pair of tangents be drawn to an ellipse,
the chords of contact will pass through a
fixed point.'

I will not trouble you with the proof of
this proposition, as it is evident to all
mathematicians, and can easily be demon-
strated. But mark well the deductions,
when we interpret this mathematical lan-
guage in correct polemical terms. A State,
through various convulsions of its own,

4—2

52 The Romance of Mathematics.

has merged into a condition represented
by a straight line, having lost its sym-
metry, its beauty, its curvilinear propor-
tion An individual unhappily situated
in this unfortunate community regards
with longing eyes the prosperous condition
of those who enjoy the social advantages
of a settled form of government, and other
blessings which accompany elliptical juris-
diction and laws. [Two tangents are
drawn to an ellipse.] No matter where
the individual may be in the unhappy
envious straight line, the result of his re-
flection will be the same. Sympathetic
chords are drawn, joining the points of
contact of the tangents with the curve ;
they all pass through a fixed point. All
these conclusions of the various individuals
on the straight line will be the same. All
are of opinion that the elliptical form is
the bestj and they mourn in secret over
the sad events which have occurred in
their own national life, their eccentricity,
their lawlessness, when they see the
advantages which their more staid and
sober-minded neighbours so freely enjoy.

Social Properties of a Conic Section. 53

2. The normal at any point of an ellipse
bisects the angle between the focal dis-
tances of that point

The normal is the perpendicular from
the point on the major axis ; it is the line
of thought directed by the observance of
just laws and rules. Hence this proposi-
tion shows that the individual citizen,
when guided by sound judgment, regards
with equal favour and entire approval the
existence of both foci, or Houses of Legis-
lature. He considers that both are neces-
sary to his comfort, and the right regulation
of the State's welfare. He cares not for
the abnormal condition of those who talk
as if the existence of either House were
unnecessary to his country's weal, and
bestows a pitying glance on those wander-
ing lights, or disturbed erratic govern-
ments, which do not possess the advantages
which from experience he has learned to
love and to respect. No matter what his
condition may be, the same opinions are
held by all classes^ all ranks and degrees ;
and if a self-opiniated particle think other-
wise, he ought to be transferred to a less

54 The Romance of Mathematics.

enlightened sphere, and migrate to a para-
bolic state, or uninteresting straight line.
And when he has changed his location,
he will look back on his old home and
old surroundings with longing eyes and an
aching heart, thinking of the blessings he
has lost by his own rash act This can
be proved mathematically. He looks for
an ideal state of society, leaps after the
shadow his fancy has depicted ; and when
he finds himself outside his former state,
he looks back with longing eyes at the
once-scorned focus. What is the focus of
a perpendicular on the tangent of an
ellipse from any external point ? Can it
not be proved to be a drck ? That is to
say, he will be more conservative than
ever. He would like to return to a primi-
tive form of government. Farewell to his
wild schemes and revolutionary measures !
Farewell to his disestablishments, aboli-
tions, and suppressions ! The throne and
government have new attractions in his
eyes; loyalty, a new feeling, asserts its
benign influence ; and if he could return

Social Properties of a Conic Section. 55

to his former position, his normal con-
duct would be straighter than ever, for
by sad experience he has learned the
value of those things which he once
despised.

But we need not depend upon one
proof alone. Exactly the same result may
be obtained from the well-known proposi-
tion which states that ' the angle between
the tangent from any external point and
the focal distance is equal to the angle
between the other tangent and the focal
distance.'

3. The same opinions are often held
by individuals in quite different walks and
classes of life. Let these individuals be
represented by points on an ellipse. Join
these, and we have a system of parallel
chords. Draw a straight line through the
middle points of these chords, and lo ! it
will always pass through the centre. This
shows that the central thought of all people
is directed to the sovereign — that loyalty
is inherent in the hearts of those who
recognise elliptical laws.

56 The Romance of Mathematics.

I will conclude this lecture with a few
remarks on the nature and properties of
the radical axis. This name was first
given, I believe, by M. Gaultier, of Tours,
and for a full account of its nature I refer
you to \ht Journal de VEcoU Polytechnique^
xvL, 1 813. The radical axis of two circles
is the line perpendicular to the line join-
ing the centres, from any point of which
the tangents to the circles are equal Let
us suppose that one circle becomes a point,
and that this point is situated on the cir-
cumference of the first circle. What is
the result? The radical axis becomes
the tangent to the circle. Hence we
may conclude that in a social system of
monarchical government the radical axis
is perpendicular to the line attaching the
individual with the monarch. Therefore
we may conclude that the radical axis
indicates a tendency of particles, or indi-
viduals, to fly off at a tangent, at right
angles to the connecting-link between the
individual and the king. When any motion
takes place, this is evident, and this ten-

Social Properties of a Conic Section. 57

dency is called centrifugal force. Sad is
it for the State when this force is called
into play, and the radical axis is a stand-
ing menace to the stability of States and
nations. The only way to counteract its
baneful, disturbing influence is to increase
the attraction of the monarch on the indi-
vidual, which nullifies the former force,
and prevents further mischief. This is the
method which nature itself adopts in the
motions of the planetary worlds ; the at-
traction of the sun prevents any disturb-
ance which might be caused in the course
of the planets by the action of centrifugal
force, and nature suggests this plan for
our adoption. Increase the attraction of
the Throne; rigidly connect each individual
by the strong chords of affection, advan-
tage and utility with the ruling power;
and then, though the radical axis may be
there, it will cease to indicate any motion
along it, it will not prevail over the counter-
acting influence of loyalty, and the stability
of the social system and the happiness of
the individuals will be the results*

58 The Romance of Mathematics.

' I would serve my King,
Serve him with all my fortune here at home,
And serve him with my person in the wars ;
Watch for him, fight for him, bleed for him, die for

him,
As every true-bom subject ought.'

This, most noble professors, is the lan-
guage of true patriotic loyalty. Let the
monarch be loved and loving, let the
laws be just and equal, happy will be the
people, prosperous the realm. There are
those who counsel different things, and
preach sedition and the breaking-up of
lawsj but those who advocate such doc-
trines lack that judicial mathematical
training which we, students and professors
of Girtham College, have acquired It
polemical mathematics, the science of the
future, should become more widely studied;
if its results were disseminated far and
wide; above all, if the proper position
which women ought to occupy in the
counsels of the nation were assigned to
them, we should hear less of these wild
schemes and foolish theories, and the in-
fluence of women would tend greatly to

Social Properties of a Conic Section. 59

promote the stability and security of the
State.

Why, let me ask, should woman be
excluded from that position which is so
justly hers ? from those duties which she
can discharge so faithfully ? It has been
said that if we wish to know the political
and moral condition of a State, we must
ask what rank women hold in it We are
told that women have more strength in
their looks than men have in their laws.
Why, then, do men debar her from those
fields of occupation wherein she may
labour for the nation's good, and use her
influence, which they acknowledge to be
great, in those callings wherein she may
most easily benefit the State, and the
country she so ardently loves ?

At some future time I hope to speak
more fully on this subject; and in con-
cluding this lecture, I will remark that
English politics need a leavening influence
which will counteract the evil tendencies
and corrupt theories which, in spite of our
advantageous social system, at present

6o The Romance of Mathematics.

exist ; and this leavening influence will be
best produced by the admission of those
into the counsels of the nation who are
acknowledged to have a benign and healthy
influence — the women of England. Let
women have their proper share in the
government of the country, and I have
no fear lest we shall preserve our elliptical
constitution, and all the advantages which
we at present enjoy.

[Editorial Note.] — In the bundle of
papers which contained the foregoing
lectures^ some letters of great interest
were found, which show that the fame
of the learned Lady Professor of Girtham
College had already gone abroad, and
attracted the attention of the leading
statesmen of the day. It is to be re-
gretted that the answers to these letters
are not forthcoming, as it might be proved
from them that the science of polemical
mathematics has already influenced the
minds of our legislators in their conduct

Social Properties of a Conic Section, 6i

of affairs at home and abroad. The
following letter is of unique interest, and
may be taken as evidence of the favour-
able impression which this new science
has made on the mind of one of our
greatest thinkers and statesmen :

Downing Street,

May, i8—

My dear Lady Professor, — The
report of the amazing results of your
scientific researches has reached me, and
I congratulate you most heartily on the
originality and acumen which you have
displayed in your investigations. A new
light has dawned upon our country.
Instead of groping in the darkness of
political warfare, ensnared by party ties
and jealousies, the statesmen of the future
will be able to calculate and determine the.
correct course with mathematical precision
and perfect accuracy. No one can
dispute the truth of a proposition in
Euclid, or the genuineness of Newton's
laws ; and if your method enables men to

62 The Romance of Mathematics.

calculate and determine the correct poli-
tical course of action, to solve political
problems as easily as exponential equa-
tions, why — then adieu to the bickerings
of party, the querulous complaints of the
Opposition I Nay, joy to the Ministry !
There will be no Opposition ! Our states-
men will be able to guide the great ship of
the State by means of charts which know
no error; and they will resemble an
association of savants met together to
determine the exact moment of the
transit of Venus, or to examine the degree
of density of a comet's tail

This condition of Parliamentary pro-
cedure is much to be desired; you have
shown how such an ideal state of things
may be obtained. In the name of the
Cxovemment I thank you for your en-
deavours on behalf of your country's
welfare, and look forward to a further
development of your admirably conceived
system. As in the domain of ordinary
science there are complex questions
which defy the acumen of the philosopher ;

Social Properties of a Conic Section, 63

so in polemical science there may be
questions which present the same difficul-
ties and complications. But as the first
are daily yielding before the persevering
attacks of the mathematician, so I doubt
not polemical science will soon over-
come the various problems which may
arise.

But it is mainly on my own account
that I venture to address you. I desire
to consult you with regard to certain
matters — political complications — which
have recently occupied the attention of
Her Majesty's Ministers. By the help of
your new science, can you aid us in our
deliberations? Of course, I am writing
to you in sfrtcf confidence^ and beg that you
will keep this communication profoundly
secret. I fear that would be a hard task
for many of your sex, who do not possess
your knowledge and powers of mind;
but I have great confidence in your dis-
cretion.

These are the problems which are
presented to us for solution :

64 The Romance of Mathematics.

1. Some members of the Cabinet are
secretly in favour of Protection, and the
country is rather stirred by the question.
Can you, from your knowledge of the
contact of curves and nations, help us to
determine what course we ought to take
with regard to Spain, for example ? Are
the principles of Adam Smith mathema-
tically correct ?

2. I observe that England is represented
mathematically by an ellipse. Are we
right in assuming that Ireland is a portion
of that ellipse? Or, on the other hand,
in our chart of nations, must we describe
that troublesome country as a rotating
parabola, or complex figure, altogether out-
side our more favoured State ?

3. Do you consider, from your minute
observation of our social system, that the
form of our elliptical government is
gradually undergoing a change, and that
a revolutionary parabolic tendency is
observable in the action of individual
particles ?

4. Is it not-possible that the differences

Social Properties of a Conic Section. 65

in the policy of the various nations of
Europe; the difficulties which beset the
carrying out of international law; the
jealousies, quarrels, and rivalries of
States, might disappear, if the same form
of government (/>., elliptical) were adopted
in each?

If you will kindly favour Her Majesty's
Ministers with your opinion on these
questions, they will owe you a debt of
gratitude, which they, as representatives of
the nation, will do their utmost to repay.

With every good wish for your further
success in the regions of polemical science,
I beg to remain.

My dear Lady Professor,
Your faithful servant,

[Editorial Note. — The next letter
is not of quite the same pleasing nature
as the foregoing, and shows that it is

* It is to be r^etted that this letter has evi-
dently fallen into the hands of some autograph
collector, who has ruthlessly cut off the signature ;
but the reader will easily determine, after careftil pe-
rusal of the document, from whose pen it emanated.

5

66 The Romance of Mathematics.

impossible to please everyone, even if that
happy consummation were desirable. This
letter was evidently called forth by some
remarks which the learned Lady Professor
had made in her third lecture with re-
ference to eccentricity in dress. Our
readers will recollect that the professor
pointed out that an extravagant * bloomer '
costume — half male, half female — was no
more a sign of genius than aesthetic
dresses always betokened the artist*
This latter statement evidently gave great
offence to the members of a society
which called itself the '^Esthetic and
Dress Improvement Association,' and
the following letter is the result of one of
their solemn conclaves :

Oscar Villa, South Kensington,
June, 1 8 — .

The Secretary of the ^Esthetic and
Dress Improvement Association presents
his compliments to the Lady Professor of
Girtham College, and begs to contradict
emphatically her statements with r^ard

• Cf. page 36.

Social Properties of a Conic Section. 67

to a subject upon which she is evidently
in entire and lamentable ignorance, and
to protest against her aspersions upon
the artistic studies of this and kindred
societies. He begs to state that true
aesthetes are nof eccentric (they leave that
to lady professors and her Philistine fol-
lowers) ; that to dress becomingly is one
of the principal objects of life, and that
true greatness is achieved as much by the
study of the art of dress as by any other
noble pursuit or graceful accomplishment
Are not Horatio Postlethwaite, Leonara
Saffronia Gillan, Vandyke Smithson en-
titled to greatness ? And yet their laurels
have been won solely by the art of dress.
Perhaps the lady professor has never read
' Sartor Resartus ' I In conclusion, he
would ask the Lady Professor to refrain
from casting obloquy upon the work of
the Association which he has the honour
to represent ] to prevail upon her pupils
to abandon the unfeminine attire which
some of them have assumed, contrary to
the first principles of art ; to array them-
selves in flowing robes of sage-green and

5—2

68 The Romance of Mathematics.

other choice colours (patterns enclosed),
and to study art, instead of absurd mathe-
matics, which no one can understand, and
do no one any good.

(Approved by the Committee of the
^Esthetic and Dress Improvement
Association.)
June, i8 — .

[Editorial Note]. — The next letter,
written by a pupil of the Lady Professor,
requires no explanation, and speaks for
itself.

Jesus Collie, Cambridge,
March, i8 — .

My dear Tutor,

You will be glad to hear that after
superhuman exertions I have at last suc-
ceeded in passing my Little-go, and I am
etemially grateful to you for all you have
done for me. I should never have got
through if it had not been for you. All
the coaches in Cambridge would never
have managed it, but you drove me
through in a canter. And why? I never
could make up my mind to work for them;

Social Properties of a Conic Section. 6g

but when I coached with you, you made
me like it I almost revelled in the
Binomial when you wrote it out for me ;
and then I could not help listening to
you ; and you looked so grieved when I
would not learn, and made me feel such
a brute ; so somehow or other you drove
some mathematics into my head, and I
pulled through. By-the-bye, I think you
must have tried the 'brain wave' dodge
with the examiners, as five out of the six
propositions in Euclid, which you told me
to get up specially, were set ! I wish I
could read people's thoughts; can you read
mine ? If I were a Don, or a Fellow, or
something, I would advise the University
to have some lady professors like you to
teach the men, instead of some of these
sleepy old tutors. It would be a great
improvement, and I am sure we should
get through a great deal more worL

They have given me a place in the Jesus
Eight, which I shall take now that I am
released from your professorial ban, and
have time for rowing. But I don't half
like giving up mathematics. You see, I

70 The Romance of Mathematics.

have grown fond of the study. Do you
think you could make a wrangler of me ?
At any rate, I should like to come to your
lectures again. May I ?

Your Grateful Pupil.

PAPER V.

A LECTURE UPON SOCIAL FORCES, WITH
SOME ACCOUNT OF POLEMICAL KINE-
MATICS.

Most noble Professors and Students
OF GiRTHAM College, — Since last *I
wandered 'twixt the pole and heavenly
hinges, 'mongst encentricals, centres, con-
centricks, circles, and epicycles,' like the
great Albumazar, and found them full of
life and wisdom for the guidance of our
States and laws, I have turned my atten-
tion to the Applied Mathematics, in order
to determine what other truths this shaft
may yield.

The strength of all sciences, according
to Bacon, consists in their harmony ; and
it is truly marvellous how perfect this
harmony is, if our ears are tuned aright to
hear it. We have observed how the

72 The Romance of Mathematics.

beautiful and regular laws of curves and
cones correspond to the social laws of
States and nations, guiding them as if by
word of counsel, admonishing them on
what principle they ought to regulate
their governments and inter-relations. We
have seen that the laws which govern
thought and light and sound are almost
identical, and that harmony pervades not
merely the ordinary sciences, but extends
her benign influence over these newly
discovered fields of scientific research,
which I claim to have discovered.

All ' this may appear at first sight sur-
prising; but the real philosopher, who
knows that all kinds of truth are inti-
mately connected, will receive such reve-
lations of science with satisfaction rather
than astonishment ; for this new science,
which has opened itself out before me, is
only an extension of other well-known
laws and discoveries which have come
down to us from the remote past

If my investigations should appear to
you, most noble professors, somewhat novel
and imaginary, remember the maxim of

Social Forces. 73

the sage, that in the infancy of science
there is no speculation which does not
merit careful examination ; and the most
remote and fanciful explanations of facts
have often been found the true ones.
Perhaps some * self-opinionated particle '
(I speak mathematically) may have been
inclined to laugh at our theories and dis-
coveries, as the wise fools of the day
laughed at Kepler and his laws ; but time
has changed the world's laughter into
praise, and a century hence our discoveries
may rank among the achievements of
modern science. As Cicero says, * Time
obliterates the fictions of opinions, but
confirms the decisions of nature.'

I have not shunned, most noble pro-
fessors, to enlist Imagination under the
banner of Geometry ; for I am fully per-
suaded that it is a powerful organ of
knowledge, and is as much needed by the
mathematician as by the poet or novelist
It is, I fear, often banished with too much
haste from the fields of intellectual re-
search by those who take upon themselves
to give laws to philosophy. We need

74 The Romance of Mathematics.

imagination to form an hypothesis; and
without hypotheses science would soon
become a lifeless and barren study, a
horse-in-the-mill affair ever strolling round
and round, unconscious of the grinding
corn. In my previous investigations my
imagination pictured the symmetry of
curves and States ; the hypothesis followed
that the laws which regulated them were
identical, and you have observed how the
supposition was confirmed by our sub-
sequent calculations.

In this lecture I propose to examine
some of the forces which exist in our
social system, and shall endeavour to
estimate them by methods of mathe-
matical procedure and analogical reason-
ing. We will begin with the old definition
of Force as that which puts matter into
motion^ or which stops^ or changes^ a
motion once commenced, When a mass
is in motion, it has a capacity for doing
work, which is called Energy; and when
this energy is caused by the motion of a
body it is called Kinetic Energy (in
mathematical language K£«^ MV^).

Social Forces. 75

Another form of kinetic energy is called
Potential Energy, which is in reality the
capacity of a body for doing work owing
to its position. For example we may take
an ordinary eight-day clock. When the
weights are wound up, they have a certain
amount of potential energy stored up,
which will counteract the friction of the
wheels and the resistance of the air on
the pendulum. Or, again, we have the
example of a water-wheel : first the water
in the reservoir, being higher than the
wheel, has an amount of potential energy.
This is converted into kinetic energy in
striking against the paddles, and after this
we have potential energy again produced
by the action of the fly-wheel.

By the principle of conservation of
energy, if we consider the whole universe,
not our planet alone (for its heat and
energy are continually diminished to some
slight degree), we find that no energy is lost.

Force is recognised as acting in two
ways : in Statics^ so as to compel rest, or
to prevent change of motion; and in
Kinetics, so as to produce or to change

76 The Romance of Mathematics.

motion ; and the whole science which in-
vestigates the action of force is called
Dynamics.

All this is of course pure mathematics,
and I have made these elementary obser-
vations for the benefit of my younger
hearers, the students of this University.
My grave and reverend seniors will pardon,
I am sure, the repetition of facts well
known to them for the sake of those who
are less informed than themselves.

Now before I proceed further, I will
endeavour to point out that these elemen-
tary truths of physical science hold good
in our social system. Each individual is
a mass, acted on by numerous forces,
capable of * doing work,' which work can
be measured and his velocity calculated.
Some individuals have a vast potential
energy; that is to say, from their position
and station in the social system, they have
a power which is capable of producing
work which a less exalted individual has
not Like the weights in an eight-day
clock, or the water in a reservoir, they
have a capacity for doing work, owing to

Social Forces. 77

the position to which they have been
raised. How vast the influence of a
Primate or a Premier, a General or a
King ! And yet their power is chiefly
potential energy, arising from the position
they occupy, not from the individuals
themselves. Schiller has described this in
poetical language, which, strange to say, is
mathematically correct :

* Yes, there's a patent of nobility
Above the meanness of our common state ;

With what they do the vulgar natures buy

Their titles ; and with what they are, the greats

Other forces may have raised these men
to their exalted positions ; but their influ-
ence is due to their height, their potential
energy. Placed on a lower level, they
would cease to have that power. How
calm the dignity of this potential rank !
The water in the reservoir is scarcely
ruflled or disturbed, as if unconscious of
its power; when it has lost its force it
rushes along with a sullen murmur and a
roar, howling and hissing and boiling in
endless torture, until —

' It gains a safer bed, and steals at last
Along the mazes of the quiet vale.'.

78 The Rofnance of Mathematics.

So the vulgar crowd rushes on, with
plenty of kinetic force, making noise
enough and looking very busy; while
those who seem to sleep in calm forget-
fulness, exercise their potential energy, and
do the real work of turning the great
engine of the State.

There are attractive and repulsive forces
(more commonly the latter, the cynic will
say) in our social system, but each indi-
vidual is the centre of various forces acting
upon him. In nature all matter possesses
the force of gravity, and whatever the
size of two particles may be, they mutually
attract each other. The earth attracts the
moon ; the moon attracts the earth. A
stone thrown up into the air exercises an
infinitesimal force upon the earth ; so in
the social system every individual, how-
ever small and insignificant he may be,
exercises some attractive force upon his
neighbour. There is no one in the world
who does not exercise some influence for
good or for evil upon his fellows.

The force of cohesion is manifest in
society as in nature, that force^ I mean.

Social Forces. 79

which resists the separation of a body's
particles. Different bodies possess differ-
ent powers of cohesion, e.g., the cohesion
of chalk is far less than that of flint em-
bedded in it ; even the same bodypossesses
different powers of cohesion in different
directions, e,g., it is easier to split wood in
the direction of the fibres than perpen-
dicular to them. If by our old principle
of continuity we change the words 'bodies '
into * States ' or * individuals,' we shall see
that the same laws hold good in social
science as in natural philosophy.

These are a few analogous laws which I
have taken almost at random ; but it must
strike the most casual listener to my
remarks that it is wondrous strange that
men, regarded as social beings, should
possess the same qualities, and be governed
by the same laws, as the rest of matter. As
Bishop Butler says, * the force of analogy
consists in the frequency of the supposed
analogous facts, and the real resemblance
of the things compared.' It appeals to
the reasoning faculty, and may form a solid
argument Hence, if we can prove the

8o The Romance of Mathematics.

similarity of various laws and conditions, we
may not be wrong in assuming by analogy
the identity of those laws and conditions.

I have stated my case in this manner in
order to convince the gainsayers, if any
such there be, and to banish any doubts
or questionings which may have arisen
in your minds. I will now proceed with
some further investigations, full of the
most profound interest and importance.

Doubtless many of the lady-students
present are in the habit of welcoming
peaceful evening in with a potent draught
of 'the cup which cheers but not
inebriates ; ' and as men are great
flatterers (for imitation is the greatest
flattery), I believe the male portion of my
audience have been known to follow that
excellent example. Some perhaps are in
the habit of burning the midnight oil, and
keep their eyes open by means of this
fruit of the hermit's pious zeal, endowed
by high omnipotence with the power of
hindering sleep ;* but that practice I do

* A Chinese l^end relates that a pious hermit,
who in his watchings and prayers had often been

Social Forces. 8t

not advise, as that delicate portion of our
system, the nerves, especially of women,
often becomes injured by such stimulating
doses. However, you will have observed
(if you do not follow the modem perni-
cious fashion of taking tea without sugar)
that numerous bubbles are formed upon
the surface of the liquid. After a few
moments these unite into one central
mass of bubbles by the force of mutual
attraction.

It appears from considerations which
are detailed in works on physical astro-
nomy, that two particles of matter placed at
any sensible distance apart attract each other
with a force directly proportional to the pro-
duct of their masses, and inversely propor-
tional to the square of their distance.

Now, suppose that we have a number
of circular masses situated upon a plane

overtaken by sleep, so that his eyelids closed, in
holy wrath against the weakness of the flesh, cut
them off, and threw them on the ground. But a
god caused a tea-shrub to spring out of them, the
leaves of which exhibit the form of an eyelid bor-
dered with lashes, and possess the gift of hindering
sleep. — Dr. Ure.

6

82 The Romance of Mathematics.

surface, they will attract each other with a
force which may be determined with
exactitude; and the greater the masses
the greater the force. We will now apply
this to polemical science. The agricul-
tural settlement is the first stage in the
civilization and formation of a State.
How did this arise? First, a single family
immigrated to some uncultivated parts
of the country, perhaps accompanied by
others, who formed a little colony. Other
settlements were made in other parts of
the land ; and thus the country became
overspread with these detached and sepa-
rate communities. An eminent writer
declares that these settlements can be
traced in the beginnings of every race
which has made progress ; that they were
characteristic of those races in Greece and
Italy, in Asia and Africa, which grew into
the opulent and famous cities in which so
much in the early history of civilization
was developed. The colonies of England
have been formed in the same way, just
as in olden time England itself was occu-
pied when the Roman power ceased.

Social Forces. 83

These settlements correspond to the
circular masses situated on the plane
surfe.ce; they were quite separate from
each other, each having its own laws, its
own headman or ruler, its own assembly
or parish council. But as time elapsed,
the force of mutual attraction set in;
by degrees these separate settlements
were drawn together by force which in-
creased in proportion as the settlements
increased ; until at last one united king-
dom was formed under one king, governed
by uniform laws and regulations. The
bubbles have blended, the circles have
come together, and one large curcle or
other curve is the result This may be
called the Law of Social Attraction, In
accordance with the results of one of my
previous lectures, I have taken the circle
as representing the simplest form of
government, which figure, in the case of
the elementary settlements, must have
been small.

Many of you, most noble professors,
are doubtless accustomed to make ex-
periments with the microscope. I will

6—2

84 The Romance of Mathematics,

suggest a simple one, which illustrates
very forcibly what I am endeavouring to
show you. Take some particles of copper,
and scatter them at intervals over the
surface of an object-glass, and pour some
sulphuric acid upon the glass. Now, what
is the result? A beautiful network of
apparently golden texture spreads itself
gradually over the whole area of the
glass. Steadily it pursues its way, and
the result is beautiful to behold. The
minute particles of copper were the origi-
nal settlements scattered over the land;
the sulphuric acid the civilizing agent;
and the final picture of a united civilized
homogeneous nation is well represented
by the progressive and finally glorious
network of gold. This example is of
course outside our present subject, but it
serves as a beautiful illustration.

As an instance of the attractive force
exercised by small communities upon each
other, I may mention the united kingdom
of Germany, which is composed of nume-
rous small States and nations, which have
been drawn together by the power of

Social Forces. 85

mutual attraction. Until recently they
were each self-contained, separate consti-
tutions, with their own kings and forms of
government; but the attracting force,
assisted by forces from without, has
proved too much for them, and the great
and ^powerful united kingdom of Germany
is the result

But why, you may ask, have not the
people in Hindustan united in the same
way ? There the agricultural settlements
remain as they did ages ago ; separate
petty chieftains rule under the all-govern-
ing power of England. Why have they
not united ?

To this objection I reply that there is
in social science, as in Nature, a vis inertia;
that is to say, there is a tendency in
matter to remain at rest if unmoved by
any external agency, and also of persisting
to move, after it has once been set in
motion. The vis inetiar of some bodies
is greater than that of others, and depends
upon their weight and density. Now it
so happens that the moral vis inertia of
the Hindustani is very great, hence their

88 The Romance of Mathematics.

greatness of the State, or drag it down
with irresistible strength from its pinnacle
of glory to an abyss of d^radation ; to
estimate such forces is the great and noble
object of our lectures and researches in
this University. Prosecute, most noble
professors, your studies in this direction
with all the energy of your enlightened
intellects, and there is yet hope that this
new science, which I have endeavoured to
sketch out, however feebly, may be the
means of saving our beloved nation from
degradation and ruin, and raising her to
a higher level of glory and honour. I
hope to continue the subject of social
forces in my next lecture.

PAPER VI.

ON SOCIAL FORCES {continued) — POLEMICAL

statics and dynamics.

Most Noble Professors and Students
OF GiRTHAM, — ^We have embarked upon a
stormy sea of speculation, on a voyage of
grand discovery, and the dangerous waves
of adverse criticism, and the deceptive
under-current of prejudice, often make the
steersman's lot by no means an enviable
one. But our vessel is sound and perfectly
equipped, and therefore I do not fear to
guide her across the great unknown.

It may have occurred to you that the
problems which present themselves for
solution in social science are far more
difficult and complicated than those which
arise in ordinary mathematics. That is
undoubtedly the case; but this extra
degree of difficulty is due to the fact that

go The Romance of Mathematics.

we make no assumptions; we take the
things as they really are, not as they are
assumed to be. In physical science, if
we take into consideration the resistance
of the air, the curvature of the earth, the
rigid connection which exists between
particles in the same body, and a host of
other things which are often conveniently
neglected in elementary works, how com-
plicated the various problems become !
So we must not be surprised at some of
the difficulties which occur in social
science, as nothing is neglected ; the whole
problem is before us, and having solved it
we need not make allowances for any
falsely assumed data.

It is possible that other professors of
this science may come to slightly different
conclusions to those which I have arrived
at. That is only to be expected, because
their original observations may have
slightly varied. But in physical science
allowances are made for different observers.
In astronomy, for example, we find the
value of the 'Personal Equation.' One
observer on looking through the telescope

Social Forces. gi

may take the meridian of a star rather
differently from another watcher of the
heavenly bodies, and the personal equa-
tion is used to make allowances for this
quickness, or slowness, of observation.
So in social science there must be a
personal equation too, and our object
ought to be, in the ordinary affairs of life
as well as in the higher duties of scientific
action, to make our personal equation as
small as possible. But until the old
proverb, * Quot homines^ tot sententice^ has
ceased to have any meaning, there will be
abundant need of this most useful aid to
accuracy.

The close connection which exists be-
tween social forces and material forces is
plainly shown by the doctrine of the con-
servation of energy. * This doctrine,' says
Dr. Tyndall, 'recognises in the material
universe a constant sum of power made
up of items among which the most Protean
fluctuations are incessantly going on. It
is as if the body of nature were alive, the
thrill and interchange of its energies re-
sembling those of an organism. The

92 The Romance of Mathematics^

parts of the stupendous whole shift and
change, augment and dimmish, appear and
disappear ; while the total of which they
are the 'parts remains quantitatively im-
mutable, plus accompanies minus^ gain
accompanies loss, no item varying in the
slightest degree without an absolutely
equal change of some other item in the
opposite direction.' So do the forces in
the social world ebb and flow, rise and
fall, carrying on the same universal law
which regulates the energy of material
force.

I will now proceed to enumerate some
of those forces which exercise such a
powerful influence on society.

First, let us take the force of Public
Opinion^ which seems to exercise a relent-
less sway over the minds and manners of
mea This is a very subtle and secret
force, which is most difficult to trace, and
resembles electricity in the science of
physics. We cannot see it, but are only
able to judge of its power by its results.
Its point of application is not in the indi-
vidual, but in the collection of individuals

Social Forces. 93

who make up the social system ; and it is,
in reality, the resultant of, or the com-
promise between, the various elementary
forces which make up human society.
Yes, compromise is a purely mecha^nical
affair, based on the principle of the paral-
lelogram of forces ; and as public opinion
is the result of a compromise, we may
calculate its force. For example : ' It is
required to know the state of public
opinion in the matter of politics, when
the results of a General Election show
that the Conservatives are to the Liberals
as 10 : 9.'

Let OC be the direction of the Con-
servative force.

Let OL be that of the Liberal.

Then by dafa OC : OL :: 10 : 9.

Complete the parallelogram, and join
OP.

94 The Romance of Mathematics.

Then OP represents the force of public
opinion in magnitude and direction.

N.B. — ^The direction of OL is deter-
mined by the amount of deviation of the
policy of the Liberals from that of the
Conservatives.

As in physical, so in social science,
impulsive forces sometimes act, and effec-
tually disturb our system and our calcu-
lations. Public opinion is very liable to
the action of disturbing forces. Panic is
an impulsive force, which defies the power
of the most learned professors of social
science to determine its magnitude and
direction. Some strange unforeseen catas-
trophe — the fascination caused by a brilliant
and unscrupulous orator, a cruel wrong, a
blind revenge for real or imaginary,injustice
— will sometimes rouse one element of
passion latent in the vast body of public
opinion; so that it breaks with all that
hitherto restrained and balanced it, and
precipitates society into a course of con-
duct inconsistent with its former behaviour,
and bloodshed, revolution, the breaking-
up of laws, are the terrible results of panic
or revengeful passioa

Social Forces. 95

Society is, as it were, split up by the
terrible action of such impulsive forces,
just as wood is split up by the repeated
blows of the hatchet It is, therefore, the
duty of statesmen to increase the power
or force of cohesion, to strengthen the
fibres of the State, so that the force of
such impulsive blows may not be felt, nor
disturb the continuity of the framework
of the State. If such measures had been
adopted in the neighbouring country of
France, much miser}' might have been
avoided, and the terrible revolutions which
have so frequently convulsed her social
system entirely prevented.

Friction is another disturbing element
in our calculations, and although it may
be made a useful servant, it is a bad
master in mathematics, as in polemics.
Without the aid of friction, progress would
be impossible. For example : Take the
case of a man with perfectly smooth skates
on perfectly hard, smooth ice ; he would
be unable to reach the land unless he had
provided himself with some stones, by
throwing which he would just be able to

96 The Romance of Mathematics.

get to his destination by a backward
motion. The engine would be unable to
proceed on its iron road if it were not for
friction. The same is true in polemical
science : the government of the country
would not be able to be carried on under
our present conditions if it were not (ox party
friction. But suppose it increased indefi-
nitely, party friction becomes party obstruc-
tion; and the engine of the State would no
longer proceed smoothly and evenly along
its appointed course at the rate of sixty
miles an hour, but would resemble an old-
fashioned coach, up to its axle-trees in
mud, its motion altogether stopped by the
action of party friction.

We have seen that forces have two ways
of acting: that of compelling rest and
that of producing motion. In statics
forces act so as to prevent any change
of motion, or disturb the body's original
position. In kinetics, on the contrary,
the power is recognised as actmg so as
to produce or change a body's motion.
Now, in polemical science we have these
two ways of considering the action of

Social Forces. 97

forces. There is the statical or conserva-
tive force, which compels rest, which seeks
security, stability, and peace, and is not
ardently devoted to change. It reduces
the system to equilibrium. There are, of
course, two kinds of equilibrium — stable
and unstable — according as the social
and political system is in a healthy or
unhealthy state. If a body is in stable
equilibrium, and any slight motion takes
place, the body will return immediately to
its former position ; but if in unstable, it
will decline further and further away from
its original position, and be entirely upset.
So a healthy and sound conservative equi-
librium is hot disturbed by outside forces,
and the State will resume its former
position of stability and rest when the
opposing force is withdrawn. But an un-
healthy and insecure conservatism is as
easily disturbed as an egg balanced on its
narrow end.

The kinetics of society, that is to say
the Radical way of estimating force, is
the party of motion, generally supposed
to be the 'party of progress.' It has

7

98 The Romance of Mathematics.

therefore many attractions in the eyes
of those who delight in motion, speed,
and rushing about. To run at full speed,
to feel the keen air upon one's face, to
experience the delightful sensation of
freedom of will, and limb, are joys which
cannot be denied. Such exercise is bene-
ficial to the system, bodily or political.
Motion is the life of all things; it is
characteristic of nature ; it adores nature ;
because it is an emblem and characteristic
of life. The ceaseless rolling of the ocean
waves, the swaying of the trees, the bend-
ing of the flowers, the waving of the com,
all these All us with pleasure ; whereas a
flat uninteresting plain, unrelieved by the
motion of terrestrial objects, is depressing
to the spirit. So there is much to be said
in favour of motion, and Carlyle has
defined progress as 'living movement.*
And men love this * living movement,' and
take up the Laureate's cry :

* Forward, forward, let us range,
Let the great world spin for ever down the ringing
Grooves of change.'

But, after all, there is a danger in this

V

Social Forces. 99

everlasting motion. We cannot tell
whither this progress may lead. It may
be along a safe sure road ; but perchance
a precipice may open out before us ; and
rejoicing in the acceleration of oyr
velocity, with eyes intent upon some
distant heights of glory and ambition, we
may not discover our danger until it is too
late to stop, and a terrible plunge into an
unknown abyss of turmoil and tumultuous
waves is the alarming result of an unguarded
policy of unrestrained 'progress.' I recall
to my mind the quaint words of Holmes
which aptly illustrate my contention.

* If the wild filly, " Progress," thou would'st ride,

Have young companions ever at thy side ;

But wouldst thou stride the staunch old mare,

" Success,"
Go with thine elders, though they please thee less.'

Progress and success do not always go
together hand in hand j and while motion
is essential to life, it is not always safe to
urge a country forward at too great a
speed ; and security and stability are quite
as important to the nation's life as actual
progress.

There are other impulsive forces which

7—2

98 The Romance of Mathematics.

therefore many attractions in the eyes
of those who delight in motion, speed,
and rushing about. To run at full speed,
to feel the keen air upon one's face, to
experience the delightful sensation of
freedom of will, and limb, are joys which
cannot be denied. Such exercise is bene-
ficial to the system, bodily or political.
Motion is the life of all things ; it is
characteristic of nature ; it adores nature ;
because it is an emblem and characteristic
of life. The ceaseless rolling of the ocean
waves, the swaying of the trees, the bend-
ing of the flowers, the waving of the corn,
all these fill us with pleasure ; whereas a
flat uninteresting plain, unrelieved by the
motion of terrestrial objects, is depressing
to the spirit. So there is much to be said
in favour of motion, and Carlyle has
defined progress as * living movement.'
And men love this * living movement,' and
take up the Laureate's cry :

* Forward, forward, let us range,
Let the great world spin for ever down the ringing
Grooves of change.'

But, after all, there is a danger in this

Social Forces. 99

everlasting motion. We cannot tell
whither this progress may lead. It may
be along a safe sure road ; but perchance
a precipice may open out before us ; and
rejoicing in the acceleration of oyr
velocity, with eyes intent upon some
distant heights of glory and ambition, we
may not discover our danger until it is too
late to stop, and a terrible plunge into an
unknown abyss of turmoil and tumultuous
waves is the alarming result of an unguarded
policy of unrestrained 'progress.' I recall
to my mind the quaint words of Holmes
which aptly illustrate my contention.

* If the wild filly, " Progress," thou would'st ride,

Have young companions ever at thy side ;

But wouldst thou stride the staunch old mare,

" Success,"
Go with thine elders, though they please thee less.*

Progress and success do not always go
together hand in hand j and while motion
is essential to life, it is not always safe to
urge a country forward at too great a
speed ; and security and stability are quite
as important to the nation's life as actual
progress.

There are other impulsive forces which

7—2

100 The Romance of Mathematics,

act occasionally in the sphere of politics,
and which baffle all our calculations, and
exclude scientific considerations of the
polemical problems which arise. Ambition
is such an impulsive force, and when the
rulers of the people are actuated by it,
and struggle for money, place, and power,
politics is degraded from its position as a
science, and it becomes impossible to
estimate the result of forces so generated.

In my next lecture I propose to treat
the important subject of the Laws which
govern States and Governments, and
which regulate, generate, and control the
social forces which we have seen at work
in the body poHtic.

PAPER VII.

LAWS OF POLITICAL MOTION.

Since the last time I had the honour of
addressing you on polemical matters, I
have met with a passage in the writings of
M. Auguste Comte which afforded me
much pleasure. It seemed to be the one
word for which I had been waiting, and
confirmed many of my own impressions
and speculations. He lays down two
propositions : first, that the constructive
politics of the future must be based on the
history of the past; and second, that
political science is a composite study, and
presupposes the complete apprehension of
every branch of science, beginning with
the physical, such as astronomy, and
ending with the moral, such as ethics and
sociology. M. Comte evidently does not
regard as a vain dream and imaginative

102 The Romance of Mathematics,

speculation the theory that it will be
possible for statesmen to calculate a
policy, and to determine a course of
action by purely scientific considerations.
May I entertain the hope that in this
university, where all branches of physical
science have found a home, and are
studied by most able and learned pro-
fessors, the science of politics may be
pursued under most favourable circum-
stances ? I trust that each professor will
bring before me the results of their
deliberations, and contribute to the growth
of this particular science for which our
university has already become deservedly
famous.

My presient lecture is devoted to the
important consideration of Law, At first
sight it may appear to you that the wills
and passions of mankind are so diverse
and unknowable, that it would be absurd
to suppose that they can be calculated, or
rendered amenable to any law. But Pro-
fessor Amos has pointed out that in pro-
portion as we examine history, and compare
the actions present and past of different

Laws of Political Motion, 103

nations and states, the more uniform does
human nature appear; the more calculable
the actions, sentiments, and emotions of
large masses of people. As we have
already stated, the difficulties of the study
are not likely to deter the professors of
Girtham College from the pursuit of any
particular branch of science.

A priori we might suppose from analogy
that these polemical laws existed, as there
is no department of nature which is not
governed by law. It is an essential fea-
ture in nature, and also in government
What is political economy but the study
of certain laws of nature? These were
first discovered by Adam Smith, and have
since been traced and estimated by such
men as Ricardo, the two Mills, Professor
Caimes, Jevons, and many others. More-
over, our physical constitutions are
governed by laws, which physicians have
determined, and which it is perilous to
resist. Our moral constitution is also
governed by laws, which evidently exist,
although it is difficult to find them out.
But the nation is only an assemblage of

104 ^^^ Romance of Mathematics.

individuals; and since individuals are
so governed, it is only natural to suppose
that the nation, composed of individuals,
is so constituted and controlled. And
not only is that true, but we shall see
that polemical laws are as permanent and
universal, as invariable and irreversible,
as the laws of nature which regulate the
courses of the heavenly bodies, and raise
the tides, or depress the sandstone hills.

We may notice first the preponderant
impulse observable in a nation's life in
favour of supporting existing facts and
institutions ; and every reformer has
discovered the difficulty and danger of
changing or opposing the customs and
habits of the people. As a wheel will
travel most smoothly along a well-worn
groove, whereby friction is diminished, so
there is a natural national tendency always
to run along those paths with which the
habits and customs of the people have
made them familiar. This law is nothing
else than Newton's first law of motion,
which is quite as applicable to human
masses as to lifeless matter. The tend-

■■•..^ J- >.,—*iifciii' «" "P' l i ■ ■ ii'i ■'■ii l ^lW— >tl

Laws of Political Motion. 105

ency of matter to remain at rest, if un-
moved by any external agency, and of
persisting to move after it has once been
set in motion, is a conservative tendency ;
and is as true in political science as in,
any other.

The special branch of our science,
which we may call the Biology of Politics^
shows how absolute is the domain of law
in polemical matters. The law of human
life is that men are born, grow, become
strong and vigorous, and then decay and
die. This is the law of life, to which
we must all yield an enforced obedience.
This same law is observed to be at work
in the heavenly bodies; and astronomy
shows us that planets are born, flourish,
and at length die, just as our human
bodies do. The moon is, as you may
have observed, a dead planet, such as our
earth may be some day. The same
growth and decay are also manifest in
national life. First, there is the birth of
the nation, which sometimes lies a long
time in a dormant state, and then wakes
up to life and energy. China and Russia

io6 The Romance of Mathematics.

are examples of dormant States, just
waking from a long sleep of childishness
and ignorance. The next stage is the
strong and healthy period of its existence,
which England is at present enjoying;
and then, after various stages of gradual
decline, we come to the senile period of
national life, when every energy and
faculty, every national feeling and power
of invention, are completely exhausted.
As an example of this depressing condi-
tion, we may mention Turkey and several
of the effete States of South America.
Sometimes, when life is nearly extinct in
the human body, physicians have made
use of the power of galvanism, in order to
revive the dying energies. This process
of galvanizing a State into life was tried
by Lord Palmerston and others on the
worn-out frame of Turkey. But such
attempts can only meet with partial and
transitory success ; and where the loss of
national power and faculty betokens the
senile period of the nation's existence^ it
is vain to attempt to restore its former
life and energy. The study of the biology

.■.->..' ,' -^' •••

»■•••■

Laws of Political Motion, 107

of politics presents many interesting and
important details in this special branch
of knowledge ; and I commend this part
of our subject to the special attention of
the professor of physiology. The law of
development is observable in nations as
in nature. Recent scientific discoveries
have tended to take away all ideas of
chance in the workings of nature, and have
substituted law instead of it. It would
be unscientific and incorrect to speak of
the world being formed by the * fortuitous
concourse of atoms.' So we cannot speak
of a State being generated in this manner.
Laws — economical, geographical, natural
— preside over the formation of States and
nations, and produce their further deve-
lopment.

The laws of political motion occupy
the same prominent place in our new
science as Newton's laws do in ordinary
dynamics. These are very important in
calculating the positions which various
States will occupy in the future. First,
we have the doctrine of nationality ^ which
prevented the progress of Austria into

io8 The Romance of Mathematics.

Italy, and of the Bourbons in Naples, and
produced the amalgamation of the small
German States in the great empire of
Germany. The second law of political
motion is the doctrine of the independ-
ence of all true States, and the equality
of all States to each other. This had its
growth in feudalism ; and all the chief
wars of modern times have been the result
of the efforts of nature to establish this
law of independence. The doctrine of
intervention is a modification of the pre-
ceding law, and is applicable when the
law of necessity demands its use, such as
the restoration of order after protracted
anarchy, the abolition of slave trade, etc.
The third law is the law of morality.
Just as for each man there exists a right
and a wrong; just as duty and conscience
are certain elements in his daily mo-
tion, which dictate his course of action,
although he may chose to neglect them ;
so a nation is bound by the same moral
laws which govern the individual; and
a nation errs if it transgresses them.
Christianity is the agent which has pro-

MntK »tm [»*«i»»

Laws of Political Motion. 109

duced so powerful an influence in making
men obey the dictates of conscience and
walk in the path of duty ; and I read
with thankfulness the conclusion of Mr.
Amos, that Christianity has triumphed
quite as much in moralizing secular
politics as it has in the sphere of in-
dividual life.

These are some of the principal laws of
motion which I have observed at work in
various States and nations. Inasmuch as
political science embraces, in addition to
the physical sciences, all those branches
which are contained in ethics, economics,
jurisprudence, sociology and others, the
laws of each are generally applicable to the
whole grand subject of which my lectures
treat. Other general laws may be deduced,
and have been enumerated in my previous
lectures, from the social properties of curves
and conies ; and when our researches are
complete we may hope to produce a code
of laws for the guidance of our statesmen
which may be of immense use in determin-

no The Romance of Mathematics.

ing the policies of the future. Already
there is strong evidence that the affairs of
this country are being conducted on sound
scientific principles, rather than by any
species of guess-work or haphazard con-
trivances. The use of history is recog-
nised as extremely important in determin-
ing a future line of conduct ; and statesmen
are in the habit of endeavouring to find
from their study of the past what is the
logical sequence of events. Just as mathe-
maticians endeavour to determine the law
of a series of figures, and having found the
law, can write down the next, and the next,
ad infinitum; so scientific politicians may
be able soon to establish the various laws
of a series of events, and calculate their
course of actions. That there is consider-
able progress in this direction is manifest
by the value which they place upon statis-
tics, and their continued use of this im-
portant information.

There are a few great evils in our present
system which are strongly opposed to any
scientific methods in politics ; and in the
interests of the country as well as those of

Laws of Political Motion, iii

science they ought to be removed. One
great evil is the want of political and
scientific knowledge on the part of the
electors, who are in the habit of choosing
their representatives on personal grounds,
or party considerations, rather than on
sound principles of political science. All
this is opposed to any idea of law. Owing
to the ignorance of the electors they fall an
easy prey to adventurers and unprincipled
politicians, who make all kinds of specious
promises, tempt them with all manner of
baits, and make self-interest instead of the
welfare of the State the principle of voting.
Selfishness is the ruin of social life and in-
tercourse, the destroyer of all happiness,
peace, and mutual trust in family life or in
society. It is the root of most of the faults,
vices, and crimes in the individual ; and
who can tell the endless disasters which
will befall the State, where selfishness is the
chief motive-power of the electors and the
elected ? A selfish statesman, one who goes
into Parliament to gain his own ends and
forward his own personal interests, is a dis-
grace to society —

112 The Romance of Mathematics,

* Feeling himself, his own low self, the whole,
When he by sacred sympathy might make
The whole one self. Self, that no alien knows !
Self, far diffused as fancy's wing can travel !
Self, spreading still, oblivious of its own.
Yet all of all possessing !'

I have said that the ignorance of the
electorate makes them an easy prey to such
men ; and until they have learnt to detect
the false from the true, until they become
acquainted with the elements of political
science, and have been taught that their
own selfish interests are not the highest
aims of social government, it is vain to
hope for a reasonable method of regu-
lating the affairs of the nation, based upon
logical laws and scientific principles.

And how is this work of educating the
electors to be accomplished? Not, I
maintain, by furious speeches and rhe-
torical displays ; not by bribery, baits and
banter; but by patient, never-ceasing
labour, by lectures on history and science,
by individual instruction, is the great work
to be accomplished upon which the security
and stability of the country depend.

Then we may hope that the * Reign of

Laws of Political Motion. 113

Law' in polemical science may be ushered
in with the joyful acclamations of an en-
lightened and united people, and its benign
influence extend from the throne of the
monarch and the council-chamber of his
ministers to the hearth of the cottager.
Politicians will rule by law ; policies be cal-
culated by laws ; people vote by law; and
then methinks I see in my mind (to use
the words of the blind old poet) a noble
and puissant nation rousing herself like a
strong man after sleep, and shaking her in-
vincible locks ; methinks I see her as an
eagle, renewing her mighty youth, and kind-
ling her undazzled eyes at the full mid-day
beam; purging and unsealing her long-
abused sight at the fountain itself of
heavenly radiance ; while the whole noise
of timorous and flocking birds flutter about
amazed at what she means. Such is the
glorious vision of the * Reign of Law.' Let
it be the business of every Englishman and
Englishwoman to arrange the framework of
our social and political system, that law
may have an uninterrupted sway; then
shall we be a united, prosperous, and con-

8

114 ^^ Romance of Mathematics.

tented people, and the reign of lawless
agitators, bribery-mongers, and counterfeit
statesmen will have passed away into the
oblivion and obscurity of a more suitable
but less £3ivoured region.

PAPER VIII.

ON THE PRINCIPLE OF POLEMICAL CO-
HESION.

In my previous lectures I have had occa-
sion to mention the principle of cohesion;
but it plays so vital a part in the constitu-
tion of States and their relations to each
other that I consider it advisable to devote
this lecture entirely to it

This is a large and comprehensive
subject, and embraces such principles as
the Centralization of States; the Co-
operation of States; Monogamic Mar-
riage; Unions; Free Trade, and many
others equally important We have
already noticed that cohesion is a well-
knowi) property of matter; that its
influence is not confined to the regions of
physical sciences; and that it is the

8—2

ii6 The Romance of Mathematics.

manifest duty of all governments to increase
the force of cohesion.

Various methods have been tried to
accomplish this purpose. The principle
of Feudalism was one of the earliest
attempts to produce the cohesion of the
nation ; and, in an elementary condition of
society, it was partly successful. The
theories of * Divine Right ' and ' Social
Contract ' were other methods which have
been adopted; and the unity of the
Christian Church has been the great
means of producing the cohesion of the
State in olden times ; and its aid may be
again required for the same beneficent
object in future complications and social
disruptions.

But it is always advantageous in scientific
pursuits to go back to first principles ; and
we will adopt that method in our present
investigations. The social unit is the
family; the multiplication of families
makes the tribe; the multiplication of
tribes makes the State ; and, therefore, we
shall not be far wrong if we consider the
family tie as the first principle of political

Principle of Polemical Cohesion. 117

cohesion. I am in agreement with several
learned thinkers upon this subject when
I say that marriage is a most important
political factor; and as marriage cannot
take place without women, it is evident
that women play a very important part in
promoting-the cohesion of the State.

This prominent position was duly
assigned to women by one of our greatest
political philosophers, M. Auguste Comte,
who strongly opposed the fatal fallacy of
ancient political systems, which greatly
overestimated the powers of men, and
depreciated those of women. If the
superiority of bodily strength be the sole
cause of greatness in political and intel-
lectual pursuits, then, most noble lords of
creation, we yield to you the palm — ^you
are our masters in this respect But if, on
the other hand, it can be shown that
physical strength is not a requisite for
great achievements in these occupations;
if the powers of endurance, elasticity,
adaptability, nervous energy, and patience
are quite as needful as mere animal
strength; then we women are quite as

ii8 The Romance of Mathematics.

capable, and indeed more capable tban
men, for achieving political greatness. In
the *good old days,' when the law of
might was right, and the sticmgest arm was
the most powerful machinery in the
government of the coontry, women were
compelled natmaUy to occupy a less pro-
minent position in the conduct of the
affairs of the nation ; and for centuries
they have been degraded by a dominating
tradition, and supposed incapable of
performing duties for which they were
mentally well suited But those militant
days are past. Animal strength and
brute force are no longer needed in the
councils of the nation ; and the time has
arrived when women should cease to be
oppressed by the disparaging illogical
deductions of former generations, and
when their assistance ought to be invoked
in the great work of promoting the
nation's welfare.

I have stated that marriage is an im-
portant political factor; and, therefore,
women have always occupied a primary,
though obscure, part in political affairs.

Principle of Polemical Cohesion. 119

The cohesion of the State has been
produced by the secret influence of
family life. But it may be asked, What
kind of marriage is most conducive to
national cohesion? This question has
been carefully and conclusively answered
by a learned scientific writer, who shows
that polygamic marriage never exists in an
advanced state, as instanced by the
history of Judaism and Mohammedanism;
that a strict form of monogamic marriage
is essential to political greatness and true
progress in civilization. The cohesion of
the State is destroyed by polygamy, and
by any system which relaxes the binding
nature of the marriage tie. 'Domestic
disorganization is a sure augury of political
disruption.'

Cohesion, the essential property of all
rightly constituted nations, is often in
danger of being lost when the State is
geographically very large, or when local
interests have greater power than the
attractive force of the central government
To obviate this evil, the method of centrali-
zation has been adopted with satisfactory

120 The Romance of Mathematics,

results, as in the case of the United States
of America, and Germany.

By this means the local authorities are
brought into close relationship with the
central head, and the centrifugal in-
fluences of independent interests and cus-
toms are counteracted by the force of
central attraction. Centralization increases
the importance of the whole body, and,
like the pendulum of a clock, regulates the
movements of the whole State. In some
cases it tends to make the government
despotic, when the local governments are
entirely under the control of the central ;
and every enactment, and scheme, and
plan checked and supervised by the chief
officers of the State. Such was the system
adopted in France by Napoleon III.
But cohesion without the enforcement of
a hard and rigid connection, a general
supervision without severe tyrannical juris-
diction, are the best methods of securing
the unity of composite States.

But the force of cohesion is evidently
at work in the nation apart from centrali-
zation. Men who have a community of

Principle of Polemical Cohesion. 121

interests unite together for the purposes of
strength and mutual assistance. They
combine for the sake of securing means
of support in sickness, and form benefit
societies, such as the Order of Oddfellows
or Foresters. This force of cohesion has
produced trade unions, and similar in-
stitutions which exist for the purpose of
protecting a common interest, and giving
expression to the concurrent opinions of
the members. These have their legitimate
use in every civilized State, in spite of
some of the disadvantages which follow in
their train. There are, of course, opposed
interests in every community: attractive
forces, which produce trade unions,
guilds, corporations, companies, and the
like; and repulsive forces, which result
from the opposed interests of employers
and employed, landlords and tenants, and
similar pairs of different classes in the
community. As time goes on, and the
State advances with it, these forces will
gain in strength; the cohesion of classes
will become greater ; association will grow
as naturally as the bubbles form on the

122 The Romance of Mathematics.

surface of our evening beverage. It is a
law of nature, and therefore cannot be
resisted. But the repulsive fprces will be
no less strong, and to calculate the
resultant of these contending interests will
be the problem for practical statesmen to
solve.

The force of cohesion is also evidently
at work, not only in individual States, but
also amongst the nations of Europe, and
of the world. That is to say, there is an
evident desire for co-operation on the part
of those nations who have attained to the
highest degree of civilization and internal
cohesion. International law is based on
the principle of cohesion, and every day it
is gaining power and favour in the eyes of
our leading statesmen. The doctrine of
Free Trade, which, if universally adopted,
would be of the greatest service to man-
kind, results from a desire for co operation ;
and whatever evils may result from one-
sided Free Trade in this country at the
present time, there can be no doubt that
ultimately the complete system will be
adopted.

Principle of Polemical Cohesion. 123

Sad is the fate of a nation when the
force of cohesion is weakened The first
revolution in France is a proof of this
assertion ; there was no cohesion, no com-
mon faith, or loyalty to the throne and
Government; and indeed the Govern-
ment, which was rotten to the core, was
hardly likely to awake any feelings of
loyalty and respect; and therefore the
social disruption which followed was only
a natural sequence of events, and was
prophesied with the accuracy with which
an astronomer can foretell an eclipse.
But that is not all ; when the cohesion of
the State is destroyed, it takes a long time
to restore the action of the force ; and, as
in the case of France, further disruption is
sure to take place.

In this lecture I have already enume-
rated some of the ways in which this force
acts; there are doubtless others which
will suggest themselves to you. But I
contend that the prosperity of the State,
and the peace of the world, depend upon
cohesion. Let this be your work, most
noble professors, to promote the action of

124 ^^^ Romance of Mathematics.

this helpful and life-giving force. Pro-
mote, as far as in you lies, the sacred
union of family life. Encourage the
generous feelings of true loyalty and
patriotism amongst the people of this
realm of England ; counsel our statesmen
with regard to the primary necessity of
national cohesion, and the advantages of
international co-operation ; and your work
will be blessed ; your names will rank with
those heroes of the sword and of the pen
who have raised our beloved country to
her present pinnacle of greatness and
prosperity ; and your memory will live in
the hearts of your grateful countrjrmen.

[Editorial Note.] — We regret to state
that the various MSS. in the sealed desk
are nearly exhausted, and are therefore
compelled to present the series of lectures
on polemical studies in an incomplete
form. But we had the good fortune to
light upon a brief diary which discloses
some interesting information with regard
to the Author's life and occupations. We
append a few extracts :

Extracts from Diary, 125

Extracts from the Author's Diary.

June $rd, — Arnold called again to-day
— the fifth time during the last fortnight !
His attention is rather overpowering, and
wastes much of my valuable time. He
says he hates science — the heathen ! — and
wants me to lecture in classics. He affirms
that mathematics are dry and hard — too
hard for women, and tend to make them
unsympathetic and critically severe. I am
afraid I was rather severe with him. But
really he is very trying, and always seems
to talk like a Greek chorus in the most
profound platitudes. Arnold is a classical
tutor at Clare College. My old pupil
is getting on famously. Poor fellow! he
seems quite oppressed with his work. But
he is making great progress, and sticks
to his books like — a student of Girtham
College !

June 4/^. — Lectured on the Scientific
Basis of Blackstone's Commentaries; after-
wards received pupils until i p.m. Really
Blanch S is more tiresome than ever.

126 The Romance of Mathematics.

It appears that she has taken up with a
young undergraduate of King's, and there
is no prospect of any improvement in her
work unless this nonsense is terminated.
How foolish some of my sex are, in spite of
their improved opportunities ! I blush for
them ! Arnold has sent me a copy of Robert
Browning's ' Belaustion/ in order to make
me like classics, and give up science.
Misguided young man ! He has written
some tolerable verses on the fly-leaf; but
I have no intention of playing Belaustion
to his ' entranced youth.' These are his
verses :

' My lady dear, if I may call you so,
For you are dearer than all else beside,
I know the love you bear to golden verse,
To golden thoughts enshrined in classic lore.
To all that's beautiful ; so here I send
Some echoes of the songs of ancient dajrs,
Attuned and chanted by an English bard,
Who fires one's old love for the rolling lines
Of youthful Hellas ; may your cultur^ ear
Receive, and gladly welcome his sweet song.
And while we revel in the poet's dream,
And hear his actors speak, we'll play our parts.
You, sweet Belaustion on the temple-steps.
Taking your captors captive by your voice ;
And I, the youth who, more entranced than all,
Was bound by fetters that he would not loose ;
And so we'll play our part. What say you, dear?*

Extracts from Diary. 127

June 6/^. — Have just seen our new
Professor of Physics, Amelia Cordial, who
is an excellent woman, and well suited for
the high office which she holds. She has
told me of the foolish conduct of Lady
Mary, who is evidently of opinion that
the professorial mantle ought to have
fallen on her shoulders. Really, this
jealousy in the ranks of the learned is
most disgraceful ; and the bickerings which
arise from disappointed ambition, the envy-
ings and silly quarrels, are the weak places
in our feitiale collegiate system.

Such good news ! The wrangler list is
just out, and my hard-working pupil is
bracketed twelfth ! This is really delight-
ful, and abundantly repays us for all our
hard toil. But really I have not found
working with him distasteful ; he is such
an excellent pupil, so painstaking and
eager, that I have quite looked forward to
his coming, and found him much more
interesting than some of these foolish
maidens. But I almost dread seeing him.
He will be so elated and overpoweringly
grateful, whereas I ought to be grateful to

I2S The Rcmance of McdkeTFLxtks^

him far all his work far me ; far I am sore
he wodld neter hare gooe is far dieTnpos
if I had not persoaded him. Wen, I
wooder wbj he does not come to tell me
of his tximnph.

June 1th, — It has cocne! and I half
expected it My eag^ pcqal writes with
an the energy and knre of his noble nature
to ask me to be his wife ! He says that
IS all he cares far, and only values his
Honoors as a step to a hi^ier honom^ and
dignity, that of gaining my love and being
my husband All this is very nice to read;
but a terribly difficult problem is placed
before me for solution. I do indeed love
this dear, good fellow — ^no one could help
doing so, I am sure; but do I not love
science more? There is a stringent regu-
lation in this University that no one shall
occupy the position of professor who is
bound by any domestic ties or cares. All
married women are excluded If I say
'Yes/ I must resign my high position,
leave this beloved college, give no more
lectures to entranced audiences. In the
interests of science, ought I to refuse, and

Extracts from Diary. 129

sacrifice my heart's affections for the cause
of mathematics? But if I say 'No,* I
must give up — /iim; sacrifice his happi-
ness too, and blight his life. Was ever
anyone so perplexed ? Science, aid thine
obedient servant ! May I not determine
this vital question by thine all-pervading

light? ....

♦ ♦ ♦ ♦ ♦

[Editorial Note.] — We had just ar-
rived at this exciting moment in the life
of the learned and accomplished lady
whose writings form the subject of these
pages — a moment when love and science
were trembling in the balance — when a
footstep was heard upon the stairs leading
to our study, and ere we could secrete our
MS. the door was opened, and a well-
known voice exclaimed :

* I do not know why you should have
become so studious lately, Ernest, and
why you should refuse to take me into
your confidence. You spend hours and
hours in this room all by yourself, writing
away, and never say a word to me about
the subject of your literary work. There

9

130 The Romance of Mathematics.

was a time when things were different, and
you were not so slow in availing yourself
of my help, and asking my advice.'

We murmured something about taking
up the pen which had been laid aside by
a far abler hand, and our deep gratitude
for past assistance in our work, which could
never be forgotten.

'And do you think that I cannot help
you now?* our visitor replied, in a very
injured tone of voice. * Is the old power
dead, because it has not recently been
used ? Ernest, I think you very ungrate-
ful not to confide in me. Come, tell me
what you are writing.'

A suggestion about the proverbial
curiosity of women rose to our lips, but
died away without utterance. In the
meantime, her eyes wandered over our
study-table strewed with papers, and lighted
upon the well-worn desk.

'Why, Ernest, where did you find this?
My dear old desk, which has been lost
ever so long ! I do believe you have been
ransacking its contents! Why did you

Conclmion. 131

not tell me that you had found it? What
are you doing with my papers, sir?

The mischief was out! We tried to
explain that the world ought not to be
deprived of that which would benefit man-
kind; that the peace and prosperity of
the country might be sacrificed if it were
deprived of these discoveries of science,
which were calculated to secure such bene-
ficial results.

At length we gained our point, and ob-
tained the full sanction of the late Lady
Professor of Girtham College to publish
her papers. Thus her obedient pupil is
enabled to repay his late instructress for
all her kindness to him, and in some
measure to compensate the scientific and
political world for the loss of one of its
most original investigators in the regions
of polemical studies, which, not without a
struggle, she resigned when she deigned
to become his wife.

THE END.
Elliot Stock f Paternoster Row, London^

1

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